%I #66 Apr 15 2024 12:37:03

%S 1,1,1,1,2,2,1,3,4,2,1,4,7,6,3,1,5,11,13,9,3,1,6,16,24,22,12,4,1,7,22,

%T 40,46,34,16,4,1,8,29,62,86,80,50,20,5,1,9,37,91,148,166,130,70,25,5,

%U 1,10,46,128,239,314,296,200,95,30,6,1,11,56,174,367,553,610,496,295,125

%N Pascal-like triangle associated with A000670.

%C From _Johannes W. Meijer_, Jul 20 2011: (Start)

%C The triangle sums, see A180662 for their definitions, link this "Races with Ties" triangle with several sequences, see the crossrefs. Observe that the Kn4 sums lead to the golden rectangle numbers A001654 and that the Fi1 and Fi2 sums lead to the Jacobsthal sequence A001045.

%C The series expansion of G(x, y) = 1/((y*x-1)*(y*x+1)*((y+1)*x-1)) as function of x leads to this sequence, see the second Maple program. (End)

%C T(2n,k) = the number of hatted frog arrangements with k frogs on the 2xn grid. See the linked paper "Frogs, hats and common subsequences". - _Chris Cox_, Apr 12 2024

%H Vincenzo Librandi, <a href="/A035317/b035317.txt">Rows n = 0..100, flattened</a>

%H Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, <a href="https://arxiv.org/abs/2404.07285">Frogs, hats and common subsequences</a>, arXiv preprint arXiv:2404.07285 [math.CO], 2024. See p. 28.

%H A. Hlavác, M. Marvan, <a href="https://arxiv.org/abs/1602.06861">Nonlocal conservation laws of the constant astigmatism equation</a>, arXiv preprint arXiv:1602.06861 [nlin.SI], 2016.

%H E. Mendelson, <a href="http://www.jstor.org/stable/2690085">Races with Ties</a>, Math. Mag. 55 (1982), 170-175.

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n,k) = Sum_{j=0..floor(n/2)} binomial(n-2j, k-2j). - _Paul Barry_, Feb 11 2003

%F From _Johannes W. Meijer_, Jul 20 2011: (Start)

%F T(n, k) = Sum_{i=0..k}((-1)^(i+k) * binomial(i+n-k+1,i)). (Mendelson)

%F T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = floor(n/2) + 1. (Mendelson)

%F Sum_{k = 0..n}((-1)^k * (n-k+1)^n * T(n, k)) = A000670(n). (Mendelson)

%F T(n, n-k) = A128176(n, k); T(n+k, n-k) = A158909(n, k); T(2*n-k, k) = A092879(n, k). (End)

%F T(2*n+1,n) = A014301(n+1); T(2*n+1,n+1) = A026641(n+1). - _Reinhard Zumkeller_, Jul 19 2012

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 3, 4, 2;

%e 1, 4, 7, 6, 3;

%e 1, 5, 11, 13, 9, 3;

%e 1, 6, 16, 24, 22, 12, 4;

%e 1, 7, 22, 40, 46, 34, 16, 4;

%e 1, 8, 29, 62, 86, 80, 50, 20, 5;

%e 1, 9, 37, 91, 148, 166, 130, 70, 25, 5;

%e 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6;

%e ...

%p A035317 := proc(n,k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(A035317(n,k), k=0..n), n=0..10); # _Johannes W. Meijer_, Jul 20 2011

%p A035317 := proc(n,k): coeff(coeftayl(1/((y*x-1)*(y*x+1)*((y+1)*x-1)), x=0, n), y, k) end: seq(seq(A035317(n,k), k=0..n), n=0..10); # _Johannes W. Meijer_, Jul 20 2011

%t t[n_, k_] := (-1)^k*(((-1)^k*(n+2)!*Hypergeometric2F1[1, n+3, k+2, -1])/((k+1)!*(n-k+1)!) + 2^(k-n-2)); Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* _Jean-François Alcover_, Dec 14 2011, after _Johannes W. Meijer_ *)

%o (Haskell)

%o a035317 n k = a035317_tabl !! n !! k

%o a035317_row n = a035317_tabl !! n

%o a035317_tabl = map snd $ iterate f (0, [1]) where

%o f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i]))

%o -- _Reinhard Zumkeller_, Jul 09 2012

%o (PARI) {T(n,k)=if(n==k,(n+2)\2,if(k==0,1,if(n>k,T(n-1,k-1)+T(n-1,k))))}

%o for(n=0,12,for(k=0,n,print1(T(n,k),","));print("")) \\ _Paul D. Hanna_, Jul 18 2012

%o (Sage)

%o def A035317_row(n):

%o @cached_function

%o def prec(n, k):

%o if k==n: return 1

%o if k==0: return 0

%o return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))

%o return [(-1)^k*prec(n+2, k) for k in (1..n)]

%o for n in (1..11): print(A035317_row(n)) # _Peter Luschny_, Mar 16 2016

%Y Row sums are A000975, diagonal sums are A080239.

%Y Central terms are A014300.

%Y Similar to the triangles A059259, A080242, A108561, A112555.

%Y Cf. A059260.

%Y Triangle sums (see the comments): A000975 (Row1), A059841 (Row2), A080239 (Kn11), A052952 (Kn21), A129696 (Kn22), A001906 (Kn3), A001654 (Kn4), A001045 (Fi1, Fi2), A023435 (Ca2), Gi2 (A193146), A190525 (Ze2), A193147 (Ze3), A181532 (Ze4). - _Johannes W. Meijer_, Jul 20 2011

%Y Cf. A181971.

%K nonn,easy,tabl,nice

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_