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 A156885 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n!, read by antidiagonals. 5
 1, 1, 1, 1, 1, 2, 1, 1, 13, 6, 1, 1, 145, 2041, 24, 1, 1, 721, 3027745, 3847285, 120, 1, 1, 2401, 374286241, 9104020469425, 87029433985, 720, 1, 1, 6301, 13835524801, 139895890728482161, 3941936722370875247425, 23624400943530205, 5040, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS G. C. Greubel, Antidiagonal rows n = 0..25, flattened FORMULA T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n! (square array). T(n, k) = ( Product_{j=1..n} ((k^2*(k+1)^2*(k+2))^j -1) )/(k^2*(k+1)^2*(k+2) -1)^n with T(n, 0) = n! (square array). - G. C. Greubel, Jun 14 2021 EXAMPLE Square array begins as: 1, 1, 1, 1, ...; 1, 1, 1, 1, ...; 2, 13, 145, 721, ...; 6, 2041, 3027745, 374286241, ...; 24, 3847285, 9104020469425, 139895890728482161, ...; Antidiagonal triangle begins as: 1; 1, 1; 1, 1, 2; 1, 1, 13, 6; 1, 1, 145, 2041, 24; 1, 1, 721, 3027745, 3847285, 120; 1, 1, 2401, 374286241, 9104020469425, 87029433985, 720; ... MATHEMATICA (* First program *) T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^5 -(m+1)^4 -(m+1)^3 +(m+1)^2)^i, {i, 0, k-1}], {k, n}]]; Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 14 2021 *) (* Second program *) T[n_, k_]= If[k==0, n!, Product[((k^2*(k+1)^2*(k+2))^j -1), {j, n}]/(k^2*(k+1)^2*(k+2) -1)^n]; Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 12 2021 *) PROG (Sage) def A156885(n, k): return factorial(n) if (k==0) else product(( (k^2*(k+1)^2*(k+2))^j -1) for j in (1..n))/(k^2*(k+1)^2*(k+2) -1)^n flatten([[A156885(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021 CROSSREFS Cf. A156881, A156882, A156883, A156888, A156889. Sequence in context: A010246 A186430 A173889 * A174718 A176291 A337514 Adjacent sequences: A156882 A156883 A156884 * A156886 A156887 A156888 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 17 2009 EXTENSIONS Edited by Joerg Arndt and G. C. Greubel, Jun 14 2021 STATUS approved

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Last modified February 20 21:52 EST 2024. Contains 370219 sequences. (Running on oeis4.)