A157118 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1, read by rows.

2, 1, 1, 1, 6, 1, 1, 27, 27, 1, 1, 88, 672, 88, 1, 1, 225, 9150, 9150, 225, 1, 1, 486, 98385, 395352, 98385, 486, 1, 1, 931, 1126951, 11748681, 11748681, 1126951, 931, 1, 1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1, 1, 2673, 201755880, 16093941435, 32251030119, 32251030119, 16093941435, 201755880, 2673, 1

Offset: 0

Author

Roger L. Bagula, Feb 23 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  2;
  1,    1;
  1,    6,        1;
  1,   27,       27,         1;
  1,   88,      672,        88,         1;
  1,  225,     9150,      9150,       225,         1;
  1,  486,    98385,    395352,     98385,       486,        1;
  1,  931,  1126951,  11748681,  11748681,   1126951,      931,    1;
  1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A001263, A157114, A157117.

Programs

Magma

A001263:= func< n,k | Binomial(n-1,k-1)*Binomial(n,k)/(n-k+1) >;
f:= func< n,k | k le n select A001263(n*k+1,n-k+1) else A001263(n*(n-k)+1, k+1) >;
A157118:= func< n,k | n eq 1 select 1 else f(n,k) + f(n,n-k) >;
[A157118(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022

Mathematica

A001263[n_, k_]:= Binomial[n-1,k-1]*Binomial[n,k]/(n-k+1);
f[n_, k_]:= If[k<=n, A001263[n*k+1,n-k+1], A001263[n*(n-k)+1,k+1]];
T[n_, k_]:= If[n==1, 1, f[n,k] + f[n,n-k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)

Sage

def A001263(n,k): return binomial(n-1,k-1)*binomial(n,k)/(n-k+1)
def f(n,k): return A001263(n*k+1,n-k+1) if (kA001263(n*(n-k)+1, k+1)
def A157118(n,k): return 1 if (n==1) else f(n,k) + f(n,n-k)
flatten([[A157118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022

Formula

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1.

T(n, n-k) = T(n, k).

Extensions

Edited by G. C. Greubel, Jan 11 2022