A156789 Irregular triangle, read by rows, T(n, k) = binomial(2*n, k)*binomial(2*k, k).

1, 1, 4, 6, 1, 8, 36, 80, 70, 1, 12, 90, 400, 1050, 1512, 924, 1, 16, 168, 1120, 4900, 14112, 25872, 27456, 12870, 1, 20, 270, 2400, 14700, 63504, 194040, 411840, 579150, 486200, 184756, 1, 24, 396, 4400, 34650, 199584, 853776, 2718144, 6370650, 10696400, 12193896, 8465184, 2704156

Offset

0,3

Comments

Row sums are A137341: {1, 11, 195, 3989, 86515, 1936881, 44241261, 1024642875, 23973456915, 565280386625, 13411044301945, ...}.

References

J. Riordan, Combinatorial Identities, Wiley, 1968, p.77.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n, k) = binomial(2*n, k)*binomial(2*k, k).

Example

Triangle begins as:
  1;
  1,  4,   6;
  1,  8,  36,   80,    70;
  1, 12,  90,  400,  1050,  1512,    924;
  1, 16, 168, 1120,  4900, 14112,  25872,  27456,  12870;
  1, 20, 270, 2400, 14700, 63504, 194040, 411840, 579150, 486200, 184756;

Maple

seq(seq( binomial(2*n, k)*binomial(2*k, k), k=0..2*n), n=0..10); # G. C. Greubel, Nov 30 2019

Mathematica

Table[Binomial[2*n, k]*Binomial[2*k, k], {n, 0, 10}, {k, 0, 2*n}]//Flatten

Prog

(PARI) T(n, k) = binomial(2*n, k)*binomial(2*k, k); \\ G. C. Greubel, Nov 30 2019
(Magma) [Binomial(2*n, k)*Binomial(2*k, k): k in [0..2*n], n in [0..10]]; // G. C. Greubel, Nov 30 2019
(Sage) [[binomial(2*n, k)*binomial(2*k, k) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Nov 30 2019
(GAP) Flat(List([0..10], n-> List([0..2*n], k->Binomial(2*n, k)*Binomial(2*k, k) ))); # G. C. Greubel, Nov 30 2019

Crossrefs

Sequence in context: A203999 A330823 A199371 * A195423 A052110 A197020

Adjacent sequences: A156786 A156787 A156788 * A156790 A156791 A156792

Keyword

nonn,tabf

Author

Roger L. Bagula, Feb 15 2009

Status

approved