OFFSET
0,4
COMMENTS
L'(n,i) are unsigned Lah numbers (cf. A008297): L'(n,i)=n!/i!*binomial(n-1,i-1) for i >= 1, L'(0,0)=1, L'(n,0)=0 for n>0. T(n,0)=A000262(n); T(n,2)=A052852(n). Row sums A052897.
Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - Paul Barry, Apr 28 2007
From Wolfdieter Lang, Jun 22 2017: (Start)
The inverse matrix T^(-1) is exponential Riordan (aka Sheffer) (e^(-x), x/(1+x)): T^(-1)(n, m) = (-1)^(n-m)*A271705(n, m).
LINKS
Muniru A Asiru, Rows n=0..50 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
FORMULA
E.g.f. for column m: (1/m!)*(x/(1-x))^m*e^(x/(x-1)), m >= 0.
From Wolfdieter Lang, Jun 22 2017: (Start)
E.g.f. for row polynomials in powers of x (e.g.f. of the triangle): exp(z/(1-z))* exp(x*z/(1-z)) (exponential Riordan).
Recurrence: T(n, 0) = Sum_{j=0} z(j)*T(n-1, j), n >= 1, with z(n) = (-1)^(n+1)*A028310(n), T(0, 0) = 1, T(n, m) = 0 n < m, T(n, m) = n*(T(n-1, m-1)/m + T(n-1, m)), n >= m >= 1 (from the z- and a-sequence, see a comment above).
Meixner type recurrence for the (monic) row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: Sum_{k=0..n-1} (-1)^k*D^(k+1)*R(n, x) = n*R(n-1, x), n >=1, R(0, x) = 1, with D = d/dx.
General Sheffer recurrence: R(n, x) = (x+1)*(1+D)^2*R(n-1, x), n >=1, R(0, x) = 1.
(End)
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of this entry; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 18 2018
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (n-1)!*binomial(n, k)*LaguerreL(n-1, 1-k, -1) with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A052897(n). (End)
EXAMPLE
n\m 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 1 1
2: 3 4 1
3: 13 21 9 1
4: 73 136 78 16 1
5: 501 1045 730 210 25 1
6: 4051 9276 7515 2720 465 36 1
7: 37633 93289 85071 36575 8015 903 49 1
8: 394353 1047376 1053724 519456 137270 20048 1596 64 1
9: 4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1
... reformatted. - Wolfdieter Lang, Jun 22 2017
E.g.f. for T(n, 2) = 1/2!*(x/(1-x))^2*e^(x/(x-1)) = 1*x^2/2 + 9*x^3/3! + 78*x^4/4! + 730*x^5/5! + 7515*x^6/6 + ...
From Wolfdieter Lang, Jun 22 2017: (Start)
The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...
T recurrence: T(3, 0) = 3*(1*T(2,0) + (1/2)*T(2, 1) + (-2/3)*T(2 ,1)) = 3*(3 + (1/2)*4 - (2/3)) = 13; T(3, 1) = 3*(T(2, 0)/1 + T(2, 1)) = 3*(3 + 4) = 21.
Meixner type recurrence for R(2, x): (D - D^2)*(3 + 4*x + x^2) = 4 + 2*x - 2 = 2*(1 + x), (D = d/dx).
General Sheffer recurrence for R(2, x): (1+x)*(1 + 2*D + D^2)*(1 + x) = (1+x)*(1 + x + 2) = 3 + 4*x + x^2. (End)
MAPLE
Lprime := proc(n, i)
if n = 0 and i = 0 then
1;
elif k = 0 then
0 ;
else
n!/i!*binomial(n-1, i-1) ;
end if;
end proc:
A059110 := proc(n, k)
add(Lprime(n, i)*binomial(i, k), i=0..n) ;
end proc: # R. J. Mathar, Mar 15 2013
MATHEMATICA
(* First program *)
lp[n_, i_] := Binomial[n-1, i-1]*n!/i!; lp[0, 0] = 1; t[n_, m_] := Sum[lp[n, i]*Binomial[i, m], {i, 0, n}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)
(* Second program *)
A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];
Table[A059110[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)
PROG
(GAP) Concatenation([1], Flat(List([1..10], n->List([0..n], m->Sum([0..n], i-> Factorial(n)/Factorial(i)*Binomial(n-1, i-1)*Binomial(i, m)))))); # Muniru A Asiru, Jul 25 2018
(Sage)
def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)
flatten([[A059110(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021
(Magma)
A059110:= func< n, k | n eq 0 select 1 else Factorial(n-1)*Binomial(n, k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;
[A059110(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021
CROSSREFS
KEYWORD
AUTHOR
Vladeta Jovovic, Jan 04 2001
STATUS
approved