A171379 Triangle, read by rows, T(n, k) = A059481(n,k)*(A059481(n,k) - 1)/2.

0, 1, 3, 3, 15, 45, 6, 45, 190, 595, 10, 105, 595, 2415, 7875, 15, 210, 1540, 7875, 31626, 106491, 21, 378, 3486, 21945, 106491, 426426, 1471470, 28, 630, 7140, 54285, 313236, 1471470, 5887596, 20701395, 36, 990, 13530, 122265, 827541, 4507503, 20701395, 82812015, 295475895

Offset

1,3

Comments

Row sums are: {0, 4, 63, 836, 11000, 147757, 2030217, 28435780, 404461170, 5824442504, ...}.

The sequence is the number of connections between figurate numbers A059481 as points page 25 Riordan.

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 25.

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

T(n,k) = binomial(n+k-1, k)*(binomial(n+k-1, k) - 1)/2.

Example

Triangle begins as:
   0;
   1,   3;
   3,  15,   45;
   6,  45,  190,   595;
  10, 105,  595,  2415,   7875;
  15, 210, 1540,  7875,  31626,  106491;
  21, 378, 3486, 21945, 106491,  426426, 1471470;
  28, 630, 7140, 54285, 313236, 1471470, 5887596, 20701395;

Maple

seq(seq( binomial(binomial(n+k-1, k), 2), k=1..n), n=1..10); # G. C. Greubel, Nov 28 2019

Mathematica

Table[Binomial[Binomial[n+k-1, k], 2], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Nov 28 2019 *)

Prog

(PARI) T(n, k) = binomial(binomial(n+k-1, k), 2); \\ G. C. Greubel, Nov 28 2019
(Magma) [Binomial(Binomial(n+k-1, k), 2): k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 28 2019
(Sage) [[binomial(binomial(n+k-1, k), 2) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 28 2019
(GAP) Flat(List([1..10], n-> List([1..n], k-> Binomial(Binomial(n+k-1, k), 2) ))); # G. C. Greubel, Nov 28 2019

Crossrefs

Cf. A059481, A143418.

Sequence in context: A126319 A165553 A056314 * A373692 A078631 A352802

Adjacent sequences: A171376 A171377 A171378 * A171380 A171381 A171382

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 07 2009

Status

approved