A176419 Triangle read by rows: T(n,k) = A022168(n,k)*n!/binomial(n,k) + 1 - n!.

1, 1, 1, 1, 4, 1, 1, 37, 37, 1, 1, 487, 1405, 487, 1, 1, 8065, 69445, 69445, 8065, 1, 1, 163081, 4467745, 13564261, 4467745, 163081, 1, 1, 3926881, 357799681, 3486035233, 3486035233, 357799681, 3926881, 1, 1, 110058481, 34357076641, 1116606260881, 3583649359297, 1116606260881, 34357076641, 110058481, 1

Offset

0,5

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

q=4;

c(n,q)=Product[1 - q^i, {i, 1, n}];

t(n,m,q)=1 - n! + n!*c(n, q)/(c[m, q)*c(n - m, q))/Binomial[n, m]

Example

Triangle begins:
  1;
  1,      1;
  1,      4,       1;
  1,     37,      37,        1;
  1,    487,    1405,      487,       1;
  1,   8065,   69445,    69445,    8065,      1;
  1, 163081, 4467745, 13564261, 4467745, 163081, 1;
  ...

Mathematica

c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] = 1 - n! + n!*c[n, q]/(c[m, q]*c[n - m, q])/Binomial[n, m];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]

Prog

(PARI) T(n, q=4) = my(c=matpascal(n, q)); vector(n+1, n, vector(n, k, c[n, k]*(n-1)!/binomial(n-1, k-1) + 1 - (n-1)!));
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Nov 19 2025

Crossrefs

Cf. A007318, A022168, A176417, A176418.

Sequence in context: A158390 A386677 A228836 * A299471 A102602 A156951

Adjacent sequences: A176416 A176417 A176418 * A176420 A176421 A176422

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 17 2010

Extensions

Edited by Andrew Howroyd, Nov 19 2025

Status

approved