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URL: https://oeis.org/A178346

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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, 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
0,5
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;
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 A383753
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, May 25 2010
EXTENSIONS
Edited by G. C. Greubel, Oct 05 2024
STATUS
approved