A144405 Triangle T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1), read by rows.

1, 1, 1, 1, 18, 1, 1, 69, 69, 1, 1, 172, 606, 172, 1, 1, 345, 2890, 2890, 345, 1, 1, 606, 9885, 23580, 9885, 606, 1, 1, 973, 27321, 127365, 127365, 27321, 973, 1, 1, 1464, 65044, 523656, 1024030, 523656, 65044, 1464, 1, 1, 2097, 138636, 1770972, 5985126, 5985126, 1770972, 138636, 2097, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1).

Sum_{k=0..n} T(n, k) = A000172(n) - A000984(n) - 2^n = Hypergeometric3F2([-n, -n, -n], [1, 1], -1) - binomial(2*n, n) - 2^n. - G. C. Greubel, Mar 27 2021

Example

Triangle begins as:
  1;
  1,    1;
  1,   18,      1;
  1,   69,     69,       1;
  1,  172,    606,     172,        1;
  1,  345,   2890,    2890,      345,        1;
  1,  606,   9885,   23580,     9885,      606,        1;
  1,  973,  27321,  127365,   127365,    27321,      973,       1;
  1, 1464,  65044,  523656,  1024030,   523656,    65044,    1464,      1;
  1, 2097, 138636, 1770972,  5985126,  5985126,  1770972,  138636,   2097,    1;
  1, 2890, 271305, 5169480, 27738690, 47945268, 27738690, 5169480, 271305, 2890, 1;

Maple

A144405:= (n, k) -> binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1);
seq(seq( A144405(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 27 2021

Mathematica

Table[Table[Binomial[n, m]*(3*Binomial[n, m]^2 - Binomial[n, m] - 1), {m, 0, n}], {n, 0, 10}]; Flatten[%]

Prog

(Magma) [Binomial(n, k)*(3*Binomial(n, k)^2 - Binomial(n, k) - 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2021
(Sage) flatten([[binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 27 2021

Crossrefs

Cf. A000172, A000984, A144404.

Sequence in context: A168623 A146774 A174451 * A202671 A203004 A155497

Adjacent sequences: A144402 A144403 A144404 * A144406 A144407 A144408

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

Extensions

Edited by G. C. Greubel, Mar 27 2021

Status

approved