A035317 - OEIS

A035317

Pascal-like triangle associated with A000670.

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, 40, 46, 34, 16, 4, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 1, 11, 56, 174, 367, 553, 610, 496, 295, 125

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

From Johannes W. Meijer, Jul 20 2011: (Start)

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.

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)

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

LINKS

Vincenzo Librandi, Rows n = 0..100, flattened

Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, Frogs, hats and common subsequences, arXiv preprint arXiv:2404.07285 [math.CO], 2024. See p. 28.

A. Hlavác, M. Marvan, Nonlocal conservation laws of the constant astigmatism equation, arXiv preprint arXiv:1602.06861 [nlin.SI], 2016.

E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175.

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

T(n,k) = Sum_{j=0..floor(n/2)} binomial(n-2j, k-2j). - Paul Barry, Feb 11 2003

From Johannes W. Meijer, Jul 20 2011: (Start)

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

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)

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

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

T(2*n+1,n) = A014301(n+1); T(2*n+1,n+1) = A026641(n+1). - Reinhard Zumkeller, Jul 19 2012

EXAMPLE

Triangle begins:

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, 40, 46, 34, 16, 4;

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

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

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

...

MAPLE

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

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

MATHEMATICA

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 *)

PROG

(Haskell)

a035317 n k = a035317_tabl !! n !! k

a035317_row n = a035317_tabl !! n

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

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

-- Reinhard Zumkeller, Jul 09 2012

(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))))}

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jul 18 2012

(Sage)

def A035317_row(n):

@cached_function