A176487 Triangle read by rows: T(n,k) = binomial(n,k) + A008292(n+1,k+1) - 1.

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 71, 29, 1, 1, 61, 311, 311, 61, 1, 1, 125, 1205, 2435, 1205, 125, 1, 1, 253, 4313, 15653, 15653, 4313, 253, 1, 1, 509, 14635, 88289, 156259, 88289, 14635, 509, 1, 1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021, 1

Offset: 0

Author

Roger L. Bagula, Apr 19 2010

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   13,    13,      1;
  1,   29,    71,     29,       1;
  1,   61,   311,    311,      61,       1;
  1,  125,  1205,   2435,    1205,     125,      1;
  1,  253,  4313,  15653,   15653,    4313,    253,     1;
  1,  509, 14635,  88289,  156259,   88289,  14635,   509,    1;
  1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021,   1;
		

Links

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

Crossrefs

Cf. A000182, A007318, A008292, A033312, A036563, A141689.

Programs

Magma

A176487:= func< n, k | Binomial(n, k) + EulerianNumber(n+1, k) - 1 >;
[A176487(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 31 2024

Maple

A176487 := proc(n,k)
    binomial(n,k)+A008292(n+1,k+1)-1 ;
end proc: # R. J. Mathar, Jun 16 2015

Mathematica

Needs["Combinatorica`"];
T[n_, k_, 0]:= Binomial[n, k];
T[n_, k_, 1]:= Eulerian[1 + n, k];
T[n_, k_, q_]:= T[n,k,q] = T[n,k,q-1] + T[n,k,q-2] - 1;
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten

SageMath

# from sage.all import * # (use for Python)
from sage.combinat.combinat import eulerian_number
def A176487(n,k): return binomial(n,k) +eulerian_number(n+1,k) -1
print(flatten([[A176487(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 31 2024

Formula

T(n, k) = A007318(n,k) + A008292(n+1,k+1) - 1, 0 <= k <= n.

Sum_{k=0..n} T(n, k) = 2^n - n + A033312(n+1) (row sums).

T(n, k) = 2*A141689(n+1,k+1) - 1. - R. J. Mathar, Jan 19 2011

From G. C. Greubel, Dec 31 2024: (Start)

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

T(n, 1) = A036563(n+1).

Sum_{k=0..n} (-1)^k * T(n,k) = ((-1)^(n/2)*A000182(n/2 + 1) - 1)*(1 + (-1)^n)/2 + [n=0]. (End)