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 A130721 Sum of the cubes of the number of standard Young tableaux over all partitions of n. 2
 1, 1, 2, 10, 64, 596, 8056, 130432, 2534960, 59822884, 1718480368, 56754444440, 2110577206816, 87981286785328, 4129351961475872, 218382856010529472, 12813477368159567200, 822337333595479929044, 57213666993723455063392, 4305630141314873304140008 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The sum of the zeroth power of the number f(p) of standard Young tableaux gives the partition function (A000041), the sum of the first power of f(p) gives the involution function (A000085), the sum of the squares of f(p) gives the factorial function (A000142), so this sequence is the natural one after them. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..60 FORMULA For p a partition of n, let f(p) be the number of standard Young tableaux with shape p. Then a(n) = sum(f(p)^3) where the sum ranges over all partitions p of n. EXAMPLE a(4) = 1^3 + 3^3 + 2^3 + 3^3 + 1^3 because the five partitions of 4 (namely 4, 3+1, 2+2, 2+1+1, 1+1+1+1) have respectively 1, 3, 2, 3, 1 standard Young tableaux. MATHEMATICA h[l_] := With[{n=Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, k_, l_] := g[n, i, l, k] = If[n == 0, h[l]^k, If[i < 1, 0, g[n, i - 1, k, l] + If[i > n, 0, g[n - i, i, k, Append[l, i]]]]]; a[n_] := If[n == 0, 1, g[n, n, 3, {}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 18 2017, after Alois P. Heinz *) CROSSREFS Cf. A000041, A000085, A000142. Column k=3 of A208447. Sequence in context: A223127 A323666 A318814 * A167449 A064170 A151410 Adjacent sequences:  A130718 A130719 A130720 * A130722 A130723 A130724 KEYWORD nonn AUTHOR David A. Madore, Jul 03 2007 EXTENSIONS More terms from Alois P. Heinz, Feb 26 2012 STATUS approved

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Last modified September 18 04:33 EDT 2020. Contains 337165 sequences. (Running on oeis4.)