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

Author

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

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A000172, A000984, A144404.

Programs

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

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[%]

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

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

Extensions

Edited by G. C. Greubel, Mar 27 2021