A176021 Triangle T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1) read by rows.

1, 1, 1, 1, -10, 1, 1, 72, 72, 1, 1, -528, -678, -528, 1, 1, 4770, 6780, 6780, 4770, 1, 1, -48025, -87568, -68458, -87568, -48025, 1, 1, 524384, 1287776, 947520, 947520, 1287776, 524384, 1, 1, -6169282, -19982590, -18010942, -10305790, -18010942, -19982590, -6169282, 1

Offset

1,5

Comments

Row sums are: 1, 2, -8, 146, -1732, 23102, -339642, 5519362, -98631416, 1926628022, ...

Links

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Formula

T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1).

T(n, k) = 1 - (-1)^n*(n! + 1) + binomial(n+1, k)*( A008297(n, k) + A008297(n, n-k+1) )/(n+1). - G. C. Greubel, Feb 08 2021

Example

Triangle begins as:
  1;
  1,        1;
  1,      -10,         1;
  1,       72,        72,         1;
  1,     -528,      -678,      -528,         1;
  1,     4770,      6780,      6780,      4770,         1;
  1,   -48025,    -87568,    -68458,    -87568,    -48025,         1;
  1,   524384,   1287776,    947520,    947520,   1287776,    524384,        1;
  1, -6169282, -19982590, -18010942, -10305790, -18010942, -19982590, -6169282, 1;

Mathematica

A176013[n_, k_]:= (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];
T[n_, m_]:= 1 - (-1)^n*(n! + 1) + A176013[n, k] + A176013[n, n-k+1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten

Prog

(Sage)
def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)
flatten([[1 - (-1)^n*(factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 08 2021
(Magma)
A176013:= func< n, k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;
[1 - (-1)^n*(Factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 08 2021

Crossrefs

Cf. A008297, A176013, A176022.

Sequence in context: A142459 A157641 A129274 * A166972 A364071 A160562

Adjacent sequences: A176018 A176019 A176020 * A176022 A176023 A176024

Keyword

sign,tabl,easy,less

Author

Roger L. Bagula, Apr 06 2010

Extensions

Edited by G. C. Greubel, Feb 08 2021

Status

approved