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

Author

Roger L. Bagula, Dec 07 2009

Keywords

• nonn
• tabl

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.

Examples

			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;
		

References

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

Links

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

Crossrefs

Cf. A059481, A143418.

Programs

GAP

Flat(List([1..10], n-> List([1..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

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 *)

PARI

T(n,k) = binomial(binomial(n+k-1, k), 2); \\ 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

Formula

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