A174919 Triangle read by rows: T(n,k) = 1 + (A022167(n,k) - binomial(n,k))^2.

1, 1, 1, 1, 5, 1, 1, 101, 101, 1, 1, 1297, 15377, 1297, 1, 1, 13457, 1440001, 1440001, 13457, 1, 1, 128165, 120912017, 1146499601, 120912017, 128165, 1, 1, 1179397, 9888711365, 856987141697, 856987141697, 9888711365, 1179397, 1, 1, 10705985, 803231797825, 629770267610177, 5762806646575105, 629770267610177, 803231797825, 10705985, 1

Offset

0,5

Links

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

Formula

T(n,k) = (A022167(n,k) - A007318(n,k))^2 + 1.

T(n,k) = (A176421(n,k) - 1)^2 + 1.

Example

Triangle begins:
  1;
  1,      1;
  1,      5,         1;
  1,    101,       101,          1;
  1,   1297,     15377,       1297,         1;
  1,  13457,   1440001,    1440001,     13457,      1;
  1, 128165, 120912017, 1146499601, 120912017, 128165, 1;
  ...

Mathematica

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

Prog

(PARI) T(n, q=3) = my(c=matpascal(n, q)-matpascal(n)); vector(n+1, n, vector(n, k, 1 + c[n, k]^2));
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Nov 18 2025

Crossrefs

Cf. A007318, A022167, A174918, A174921, A176421.

Sequence in context: A174912 A106238 A173475 * A156952 A158748 A351241

Adjacent sequences: A174916 A174917 A174918 * A174920 A174921 A174922

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 02 2010

Extensions

Edited by Andrew Howroyd, Nov 18 2025

Status

approved