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A171379
Triangle, read by rows, T(n, k) = A059481(n,k)*(A059481(n,k) - 1)/2.
1
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.
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
Sequence in context: A126319 A165553 A056314 * A373692 A078631 A352802
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Dec 07 2009
STATUS
approved