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 A294746 Number A(n,k) of compositions (ordered partitions) of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals. 21
 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 10, 13, 1, 1, 1, 35, 217, 75, 1, 1, 1, 126, 4245, 8317, 525, 1, 1, 1, 462, 90376, 1239823, 487630, 4347, 1, 1, 1, 1716, 2019836, 216456376, 709097481, 40647178, 41245, 1, 1, 1, 6435, 46570140, 41175714454, 1303699790001, 701954099115, 4561368175, 441675, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Row r >= 2, is asymptotic to r^(r*n + 3/2) / (2*Pi*n)^((r-1)/2). - Vaclav Kotesovec, Sep 20 2019 LINKS Alois P. Heinz, Antidiagonals n = 0..45, flattened FORMULA A(n,k) = [x^((k+1)^n)] (Sum_{j=0..k*n+1} x^((k+1)^j))^(k*n+1) for k>0, A(n,0) = 1. EXAMPLE A(3,1) = 13: [1/4,1/4,1/4,1/4], [1/2,1/4,1/8,1/8], [1/2,1/8,1/4,1/8], [1/2,1/8,1/8,1/4], [1/4,1/2,1/8,1/8], [1/4,1/8,1/2,1/8], [1/4,1/8,1/8,1/2], [1/8,1/2,1/4,1/8], [1/8,1/2,1/8,1/4], [1/8,1/4,1/2,1/8], [1/8,1/4,1/8,1/2], [1/8,1/8,1/2,1/4], [1/8,1/8,1/4,1/2]. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, ... 1, 3, 10, 35, 126, 462, ... 1, 13, 217, 4245, 90376, 2019836, ... 1, 75, 8317, 1239823, 216456376, 41175714454, ... 1, 525, 487630, 709097481, 1303699790001, 2713420774885145, ... MAPLE b:= proc(n, r, p, k) option remember; `if`(n `if`(k=0, 1, b(k*n+1, 1, 0, k+1)): seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]]; A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, 0, k + 1]]; Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000012, A007178(n+1), A294850, A294851, A294852, A294853, A294854, A294855, A294856, A294857, A294858. Rows n=0+1, 2-10 give: A000012, A001700, A294982, A294983, A294984, A294985, A294986, A294987, A294988, A294989. Main diagonal gives: A294747. Cf. A294775. Sequence in context: A263383 A185620 A096066 * A326180 A064085 A256692 Adjacent sequences: A294743 A294744 A294745 * A294747 A294748 A294749 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Nov 07 2017 STATUS approved

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Last modified February 23 20:31 EST 2024. Contains 370288 sequences. (Running on oeis4.)