A174918 Triangle read by rows: T(n,k) = 1 + (A022166(n,k) - binomial(n,k))^2.

1, 1, 1, 1, 2, 1, 1, 17, 17, 1, 1, 122, 842, 122, 1, 1, 677, 21026, 21026, 677, 1, 1, 3250, 404497, 1890626, 404497, 3250, 1, 1, 14401, 7001317, 138674177, 138674177, 7001317, 14401, 1, 1, 61010, 115928290, 9428215802, 40287314090, 9428215802, 115928290, 61010, 1

Offset

0,5

Links

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

Formula

T(n,k) = (A022166(n,k) - A007318(n,k))^2 + 1.

T(n,k) = (A176420(n,k) - 1)^2 + 1.

Example

Triangle begins:
  1;
  1,     1;
  1,     2,       1;
  1,    17,      17,         1;
  1,   122,     842,       122,         1;
  1,   677,   21026,     21026,       677,       1;
  1,  3250,  404497,   1890626,    404497,    3250,     1;
  1, 14401, 7001317, 138674177, 138674177, 7001317, 14401, 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=2) = 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, A022166, A174919, A174921, A176420.

Sequence in context: A322621 A309036 A294756 * A154991 A090163 A179071

Adjacent sequences: A174915 A174916 A174917 * A174919 A174920 A174921

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 02 2010

Extensions

Edited by Andrew Howroyd, Nov 18 2025

Status

approved