%I #138 Jan 18 2024 05:57:18

%S 1,0,1,0,1,1,0,2,2,1,0,6,5,3,1,0,24,16,9,4,1,0,120,64,31,14,5,1,0,720,

%T 312,126,52,20,6,1,0,5040,1812,606,217,80,27,7,1,0,40320,12288,3428,

%U 1040,345,116,35,8,1,0,362880,95616,22572,5768,1661,519,161,44,9,1

%N Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

%C Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0,...] = A110654 DELTA A000007.

%C In general, the triangle [r_0,r_1,r_2,r_3,...] DELTA [s_0,s_1,s_2,s_3,...] has generating function 1/(1-(r_0*x+s_0*x*y)/(1-(r_1*x+s_1*x*y)/(1-(r_2*x+s_2*x*y)/(1-(r_3*x+s_3*x*y)/(1-...(continued fraction). See also the Formula section below.

%C T(n,k) = number of permutations on [n] that (i) contain a 132 pattern only as part of a 4132 pattern and (ii) start with n+1-k. For example, for n >= 1, T(n,1) = (n-1)! counts all (n-1)! permutations on [n] that start with n: either they avoid 132 altogether or the initial entry serves as the "4" in a 4132 pattern and T(4,3) = 3 counts 2134, 2314, 2341. - _David Callan_, Jul 20 2005

%C T(n,k) is the number of permutations on [n] that (i) contain a (scattered) 342 pattern only as part of a 1342 pattern and (ii) contain 1 in position k. For example, T(4,3) counts 3214, 4213, 4312. (It does not count, say, 2314 because 231 forms an offending 342 pattern.) - _David Callan_, Jul 20 2005

%C Riordan array (1,x*g(x)) where g(x) is the g.f. of the factorials (n!). - _Paul Barry_, Sep 25 2008

%C Modulo 2, this sequence becomes A106344.

%C T(n,k) is the number of permutations of {1,2,...,n} having k cycles such that the elements of each cycle of the permutation form an interval. - _Ran Pan_, Nov 11 2016

%C The convolution triangle of the factorial numbers. - _Peter Luschny_, Oct 09 2022

%H T. D. Noe, <a href="/A084938/b084938.txt">Rows n = 0..100 of triangle, flattened</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry2/barry73.html">A Note on a One-Parameter Family of Catalan-Like Numbers</a>, JIS 12 (2009) 09.5.4.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009), Article 09.7.6.

%H Paul Barry, <a href="https://arxiv.org/abs/2101.06713">On the inversion of Riordan arrays</a>, arXiv:2101.06713 [math.CO], 2021.

%H Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry2/barry126.html">A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations</a>, J. Int. Seq. 14 (2011), Article 11.3.8.

%H David Callan, <a href="http://arxiv.org/abs/math/0507169">A combinatorial interpretation of the eigensequence for composition</a>, arXiv:math/0507169 [math.CO], 2005.

%H David Callan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Callan/callan96.html">A Combinatorial Interpretation of the Eigensequence for Composition</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

%H H. Fuks and J. M. G. Soto, <a href="http://arxiv.org/abs/1306.1189">Exponential convergence to equilibrium in cellular automata asymptotically emulating identity</a>, arXiv preprint arXiv:1306.1189 [nlin.CG], 2013.

%H Sergey Kitaev and Philip B. Zhang, <a href="https://arxiv.org/abs/1811.07679">Distributions of mesh patterns of short lengths</a>, arXiv:1811.07679 [math.CO], 2018.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SequenceTransformations">Transformations of integer sequences</a>.

%H R. J. Mathar, <a href="/A084938/a084938.pdf">Properties of Deleham's Delta Transformation: OEIS A084938</a>

%F The operator DELTA takes two sequences r = (r_0, r_1, ...), s = (s_0, s_1, ...) and produces a triangle T(n, k), 0 <= k <= n, as follows:

%F Let q(k) = x*r_k + y*s_k for k >= 0; let P(n, k) (n >= 0, k >= -1) be defined recursively by P(0, k) = 1 for k >= 0; P(n, -1) = 0 for n >= 1; P(n, k) = P(n, k-1) + q(k)*P(n-1, k+1) for n >= 1, k >= 0. Then P(n, k) is a homogeneous polynomial in x and y of degree n and T(n, k) = coefficient of x^(n-k)*y^k in P(n, 0).

%F T(n, n) = 1.

%F T(k+1, k) = A001477(k).

%F T(k+2, k) = A000096(k).

%F T(n+1, 1) = A000142(n).

%F T(n+2, 2) = A003149(n).

%F T(n+3, 3) = A090595(n).

%F T(n+4, 4) = A090319(n).

%F T(m+n, m) = Sum_{k=0..n} A090238(n, k)*binomial(m, k).

%F G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.

%F For k>0, T(n+k, k) = Sum_{a_1 + a_2 + .. + a_k = n} (a_1)!*(a_2)!*..*(a_k)!; a_i>=0, n>=0.

%F T(n,k) = Sum_{j>=0} A075834(j)*T(n-1,k+j-1).

%F T(2n,n) = A287899(n). - _Alois P. Heinz_, Jun 02 2017

%F From _G. C. Greubel_, Nov 10 2022: (Start)

%F Sum_{k=0..n} T(n, k) = A051295(n).

%F Sum_{k=0..n} (-1)^k*T(n, k) = [n=0] - A052186(n-1)*[n>0]. (End)

%e From _Paul Barry_, Sep 25 2008: (Start)

%e Triangle [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,0,0,0,...] begins

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 2, 2, 1;

%e 0, 6, 5, 3, 1;

%e 0, 24, 16, 9, 4, 1;

%e 0, 120, 64, 31, 14, 5, 1;

%e 0, 720, 312, 126, 52, 20, 6, 1;

%e 0, 5040, 1812, 606, 217, 80, 27, 7, 1;

%e 0, 40320, 12288, 3428, 1040, 345, 116, 35, 8, 1;

%e 0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1. (End)

%e From _Paul Barry_, May 14 2009: (Start)

%e The production matrix is

%e 0, 1;

%e 0, 1, 1;

%e 0, 1, 1, 1;

%e 0, 2, 1, 1, 1;

%e 0, 7, 2, 1, 1, 1;

%e 0, 34, 7, 2, 1, 1, 1;

%e 0, 206, 34, 7, 2, 1, 1, 1;

%e which is based on A075834. (End)

%p DELTA := proc(r,s,n) local T,x,y,q,P,i,j,k,t1; T := array(0..n,0..n);

%p for i from 0 to n do q[i] := x*r[i+1]+y*s[i+1]; od: for k from 0 to n do P[0,k] := 1; od: for i from 0 to n do P[i,-1] := 0; od:

%p for i from 1 to n do for k from 0 to n do P[i,k] := sort(expand(P[i,k-1] + q[k]*P[i-1,k+1])); od: od:

%p for i from 0 to n do t1 := P[i,0]; for j from 0 to i do T[i,j] := coeff(coeff(t1,x,i-j),y,j); od: lprint( seq(T[i,j],j=0..i) ); od: end;

%p # To produce the current triangle: s3 := n->floor((n+1)/2); s4 := n->if n = 0 then 1 else 0; fi; r := [seq(s3(i),i= 0..40)]; s := [seq(s4(i),i=0..40)]; DELTA(r,s,20);

%p # Uses function PMatrix from A357368.

%p PMatrix(10, n -> factorial(n - 1)); # _Peter Luschny_, Oct 09 2022

%t a[0, 0] = 1; a[n_, k_] := a[n, k] = Sum[j! a[n - j - 1, k - 1], {j, 0, n - 1}]; Flatten[Table[a[i, j], {i, 0, 10}, {j, 0, i}]] (* _T. D. Noe_, Feb 22 2012 *)

%t DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k+1]] + y*s[[k+1]]; p[0, _] = 1; p[_, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k]*p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]]; DELTA[Floor[Range[10]/2], Prepend[Table[0, {10}], 1], 10] (* _Jean-François Alcover_, Sep 12 2013, after _Philippe Deléham_ *)

%o (Sage)

%o def delehamdelta(R, S) :

%o L = min(len(R), len(S)) + 1

%o ring = PolynomialRing(ZZ, 'x')

%o x = ring.gen()

%o A = [Rk + x*Sk for Rk, Sk in zip(R, S)]

%o C = [ring(0)] + [ring(1) for i in range(L)]

%o for k in (1..L) :

%o for n in range(k-1,0,-1) :

%o C[n] = C[n-1] + C[n+1]*A[n-1]

%o yield list(C[1])

%o def A084938_triangle(n) :

%o for row in delehamdelta([(i+1)//2 for i in (0..n)], [0^i for i in (0..n)]):

%o print(row)

%o A084938_triangle(10) # _Peter Luschny_, Jan 28 2012

%o (Magma)

%o function T(n,k) // T = A084938

%o if k lt 0 or k gt n then return 0;

%o elif n eq 0 or k eq n then return 1;

%o elif k eq 0 then return 0;

%o else return (&+[Factorial(j)*T(n-j-1,k-1): j in [0..n-1]]);

%o end if; return T;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 10 2022

%Y Cf. A003149, A051295 (row sums), A052186, A090238,

%Y Cf. A287899.

%Y Columns: A000007, A000142, A003149, A090595, A090319.

%Y Diagonals: A000012, A001477, A000096, A092286, A090386, A090391, A090392, A090393, A090394.

%K nonn,tabl

%O 0,8

%A _Philippe Deléham_, Jul 16 2003; corrections Dec 17 2008, Dec 20 2008, Feb 05 2009

%E Name edited by _Derek Orr_, May 01 2015