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%I #8 Sep 08 2022 08:45:49
%S 0,1,3,3,15,45,6,45,190,595,10,105,595,2415,7875,15,210,1540,7875,
%T 31626,106491,21,378,3486,21945,106491,426426,1471470,28,630,7140,
%U 54285,313236,1471470,5887596,20701395,36,990,13530,122265,827541,4507503,20701395,82812015,295475895
%N Triangle, read by rows, T(n, k) = A059481(n,k)*(A059481(n,k) - 1)/2.
%C Row sums are: {0, 4, 63, 836, 11000, 147757, 2030217, 28435780, 404461170, 5824442504, ...}.
%C The sequence is the number of connections between figurate numbers A059481 as points page 25 Riordan.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 25.
%H G. C. Greubel, <a href="/A171379/b171379.txt">Rows n = 1..100 of triangle, flattened</a>
%F T(n,k) = binomial(n+k-1, k)*(binomial(n+k-1, k) - 1)/2.
%e Triangle begins as:
%e 0;
%e 1, 3;
%e 3, 15, 45;
%e 6, 45, 190, 595;
%e 10, 105, 595, 2415, 7875;
%e 15, 210, 1540, 7875, 31626, 106491;
%e 21, 378, 3486, 21945, 106491, 426426, 1471470;
%e 28, 630, 7140, 54285, 313236, 1471470, 5887596, 20701395;
%p seq(seq( binomial(binomial(n+k-1, k), 2), k=1..n), n=1..10); # _G. C. Greubel_, Nov 28 2019
%t Table[Binomial[Binomial[n+k-1, k], 2], {n,10}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Nov 28 2019 *)
%o (PARI) T(n,k) = binomial(binomial(n+k-1, k), 2); \\ _G. C. Greubel_, Nov 28 2019
%o (Magma) [Binomial(Binomial(n+k-1, k), 2): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Nov 28 2019
%o (Sage) [[binomial(binomial(n+k-1, k), 2) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Nov 28 2019
%o (GAP) Flat(List([1..10], n-> List([1..n], k-> Binomial(Binomial(n+k-1, k), 2) ))); # _G. C. Greubel_, Nov 28 2019
%Y Cf. A059481, A143418.
%K nonn,tabl
%O 1,3
%A _Roger L. Bagula_, Dec 07 2009
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