A178346 - OEIS

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

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

OFFSET

0,5

LINKS

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

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

EXAMPLE

Triangle begins as:

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;

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

PROG

(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

(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

CROSSREFS

Cf. A008292.

Sequence in context: A347974 A029847 A154334 * A168551 A262307 A144397

Adjacent sequences: A178343 A178344 A178345 * A178347 A178348 A178349

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, May 25 2010

EXTENSIONS

Edited by G. C. Greubel, Oct 05 2024

STATUS

approved