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 A174696 Triangle T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1, read by rows. 3
 1, 1, 1, 1, 49, 1, 1, 841, 841, 1, 1, 11881, 47881, 11881, 1, 1, 161281, 1799281, 1799281, 161281, 1, 1, 2217601, 55560961, 154344961, 55560961, 2217601, 1, 1, 31570561, 1548892801, 9680791681, 9680791681, 1548892801, 31570561, 1, 1, 469929601, 40967337601, 501853968001, 1129171881601, 501853968001, 40967337601, 469929601, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS G. C. Greubel, Rows n = 1..100 of the triangle, flattened FORMULA T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1. From G. C. Greubel, Feb 09 2021: (Start) T(n, k) = n! * A174158(n, k) - n! + 1. Sum_{k=1..n} T(n,k) = n! * Hypergeometric4F3([-n, -n, 1-n, 1-n], [1, 2, 2], 1) - n*(n! - 1) = n! * A319743(n) - n*(n! - 1). (End) EXAMPLE Triangle begins as: 1; 1, 1; 1, 49, 1; 1, 841, 841, 1; 1, 11881, 47881, 11881, 1; 1, 161281, 1799281, 1799281, 161281, 1; 1, 2217601, 55560961, 154344961, 55560961, 2217601, 1; 1, 31570561, 1548892801, 9680791681, 9680791681, 1548892801, 31570561, 1; MATHEMATICA T[n_, m_]:= n!*(1/k)^2*(Binomial[n-1, k-1]*Binomial[n, k-1])^2 - n! + 1; Table[T[n, k], {n, 12}, {k, n}]//Flatten PROG (Sage) def A174696(n, k): return (factorial(n)/k^2)*(binomial(n-1, k-1)*binomial(n, k-1))^2 - factorial(n) + 1 flatten([[A174696(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021 (Magma) A174696:= func< n, k | (Factorial(n)/k^2)*(Binomial(n-1, k-1)*Binomial(n, k-1))^2 - Factorial(n) + 1 >; [A174696(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021 CROSSREFS Cf. A174158, A174694, A176013, A319743. Sequence in context: A115480 A214953 A005071 * A203506 A254617 A012120 Adjacent sequences: A174693 A174694 A174695 * A174697 A174698 A174699 KEYWORD nonn,tabl,easy AUTHOR Roger L. Bagula, Mar 27 2010 EXTENSIONS Edited by G. C. Greubel, Feb 09 2021 STATUS approved

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Last modified December 2 23:17 EST 2023. Contains 367526 sequences. (Running on oeis4.)