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 A156888 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^7 - Sum_{m=1..5} (k+1)^m )^i ) with T(n, 0) = n!, read by antidiagonals. 5
 1, 1, 1, 1, 1, 2, 1, 1, 67, 6, 1, 1, 1825, 296341, 24, 1, 1, 15021, 6075061825, 86507568379, 120, 1, 1, 74221, 3388969238841, 36886153511769270625, 1666711474847102245, 720, 1, 1, 270607, 408859932813241, 11484347898092358710495061, 408508286631392559053881969770625, 2119389029714451057320373595, 5040 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS G. C. Greubel, Antidiagonal rows n = 0..20, flattened FORMULA T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^7 - Sum_{m=1..5} (k+1)^m )^i ) with T(n, 0) = n! (square array). T(n, k) = ( Product_{j=1..n} (f(k)^j -1) )/(f(k) -1)^n with T(n, 0) = n! and f(k) = ((k+1)/k)*((k+1)^7 - (k+1)^6 - (k+1)^5 + 1) (square array). - G. C. Greubel, Jun 14 2021 EXAMPLE Square array begins as: 1, 1, 1, ...; 1, 1, 1, ...; 2, 67, 1825, ...; 6, 296341, 6075061825, ...; 24, 86507568379, 36886153511769270625, ...; Antidiagonal triangle begins as: 1; 1, 1; 1, 1, 2; 1, 1, 67, 6; 1, 1, 1825, 296341, 24; 1, 1, 15021, 6075061825, 86507568379, 120; 1, 1, 74221, 3388969238841, 36886153511769270625, 1666711474847102245, 720; ... MATHEMATICA (* First program *) T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^7 - Sum[(m+1)^r, {r, 1, 5}])^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 *) f[n_]:= ((n+1)/n)*((n+1)^7 - (n+1)^6 - (n+1)^5 + 1); T[n_, k_]= If[k==0, n!, Product[(f[k]^j -1), {j, n}]/(f[k]-1)^n]; Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2021 *) PROG (Sage) def f(n): return ((n+1)/n)*((n+1)^7 - (n+1)^6 - (n+1)^5 + 1) def A156888(n, k): return factorial(n) if (k==0) else product( (f(k)^j - 1) for j in (1..n))/(f(k)-1)^n flatten([[A156888(k, n-k) for k in (0..n)] for n in (0..6)]) # G. C. Greubel, Jun 14 2021 CROSSREFS Cf. A156881, A156882, A156883, A156885, A156889. Sequence in context: A362226 A058293 A172092 * A173890 A159767 A169658 Adjacent sequences: A156885 A156886 A156887 * A156889 A156890 A156891 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 August 7 03:39 EDT 2024. Contains 375008 sequences. (Running on oeis4.)