A178346 Triangle read by rows: T(n, k, m) = binomial(n, k) - m*binomial(n, k)*binomial(n+1, k)/(k+1) + m*A008292(n+1, k+1) with m = 3.

1, 1, 1, 1, 5, 1, 1, 18, 18, 1, 1, 52, 144, 52, 1, 1, 131, 766, 766, 131, 1, 1, 303, 3273, 6743, 3273, 303, 1, 1, 664, 12312, 45422, 45422, 12312, 664, 1, 1, 1406, 42844, 261230, 463348, 261230, 42844, 1406, 1, 1, 2913, 141936, 1358100, 3915312, 3915312, 1358100, 141936, 2913, 1, 1, 5953, 455481, 6595734, 29172972, 47114784, 29172972, 6595734, 455481, 5953, 1

Offset: 0

Author

Roger L. Bagula, May 25 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    5,      1;
  1,   18,     18,       1;
  1,   52,    144,      52,       1;
  1,  131,    766,     766,     131,       1;
  1,  303,   3273,    6743,    3273,     303,       1;
  1,  664,  12312,   45422,   45422,   12312,     664,      1;
  1, 1406,  42844,  261230,  463348,  261230,   42844,   1406,    1;
  1, 2913, 141936, 1358100, 3915312, 3915312, 1358100, 141936, 2913,   1;
		

Links

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

Crossrefs

Cf. A008292.

Programs

Magma

A178346:= func< n,k | Binomial(n, k) - 3*(Binomial(n, k)*Binomial(n+1, k)/(k+1)) + 3*EulerianNumber(n+1, k) >;
[A178346(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Oct 05 2024

Mathematica

EulerianNumber[n_, k_] := EulerianNumber[n, k] = Sum[(-1)^j*(k-j)^n*Binomial[n+ 1, j], {j,0,k}];
A178346[n_, k_, m_]:= Binomial[n, k] - m*Binomial[n, k]*Binomial[n+1, k]/(k+1) + m*EulerianNumber[n+1, k+1];
Table[A178346[n,k,3], {n,0,15}, {k,0,n}]//Flatten

SageMath

def A008292(n,k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in (0..k))
def A178346(n,k): return binomial(n,k) - 3*binomial(n,k)*binomial(n+1,k)/(k+1) + 3*A008292(n+1,k+1)
flatten([[A178346(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Oct 05 2024

Formula

T(n, k, m) = binomial(n, k) - m*binomial(n, k)*binomial(n+1, k)/(k+1) + m*Eulerian(n+1, k+1) with m = 3, and Eulerian(n,k) = A008292(n,k).

Sum_{k=0..n} T(n, k) = 2^n + 3*(n+1)! - 3*Catalan(n+1) = 2^n + 3*A056986(n+1). - G. C. Greubel, Oct 05 2024

Extensions

Edited by G. C. Greubel, Oct 05 2024