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 A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals. 12
 1, 1, 1, 1, 1, 3, 1, 1, 4, 10, 1, 1, 6, 27, 35, 1, 1, 10, 93, 256, 126, 1, 1, 18, 381, 2716, 3125, 462, 1, 1, 34, 1785, 36628, 127905, 46656, 1716, 1, 1, 66, 9237, 591460, 7120505, 8848236, 823543, 6435, 1, 1, 130, 51033, 11007556, 495872505, 2443835736, 844691407, 16777216, 24310 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Alois P. Heinz, Antidiagonals n = 0..50, flattened FORMULA A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n. EXAMPLE A(3,2) = 93: (z1+z2+z3)^3 = z1^3 +3*z1^2*z2 +3*z1^2*z3 +3*z1*z2^2 +6*z1*z2*z3 +3*z1*z3^2 +z2^3 +3*z2^2*z3 +3*z2*z3^2 +z3^3 => 1^2+3^2+3^2+3^2+6^2+3^2+1^2+3^2+3^2+1^2 = 93. Square array A(n,k) begins: 0 : 1, 1, 1, 1, 1, 1, ... 1 : 1, 1, 1, 1, 1, 1, ... 2 : 3, 4, 6, 10, 18, 34, ... 3 : 10, 27, 93, 381, 1785, 9237, ... 4 : 35, 256, 2716, 36628, 591460, 11007556, ... 5 : 126, 3125, 127905, 7120505, 495872505, 41262262505, ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1, add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n)) end: A:= (n, k)-> b(n\$2, k): seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] * Binomial[n, j]^(k-1)/j!, {j, 0, n}]]]; A[n_, k_] := n!*b[n, n, k]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A001700(n-1) for n>0, A000312, A033935, A055733, A055740, A246240, A246241, A246242, A246243, A246244, A246245. Rows n=0+1, 2 give: A000012, A052548. Main diagonal gives A245398. Sequence in context: A342447 A025255 A296006 * A346792 A294316 A294761 Adjacent sequences: A245394 A245395 A245396 * A245398 A245399 A245400 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 21 2014 STATUS approved

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Last modified August 14 16:54 EDT 2024. Contains 375166 sequences. (Running on oeis4.)