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 A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k. 22
 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 10, 1, 1, 1, 2, 6, 23, 36, 1, 1, 1, 2, 7, 46, 239, 202, 1, 1, 1, 2, 8, 82, 1086, 5828, 1828, 1, 1, 1, 2, 9, 134, 3707, 79326, 342383, 27338, 1, 1, 1, 2, 10, 205, 10340, 642457, 18583582, 50110484, 692004, 1, 1, 1, 2, 11, 298, 24901, 3649346, 446020582, 14481808030, 18757984046, 30251722, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..40, flattened FORMULA See program. For k>1: A(n,k) = [x^(k^n)] 1/Product_{j>=0} (1-x^(k^j)). EXAMPLE A(2,3) = 5, because there are 5 partitions of 3^2=9 into powers of 3: [1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,3], [1,1,1,3,3], [3,3,3], [9]. Square array A(n,k) begins:   1,  1,   1,    1,     1,      1,  ...   1,  1,   2,    2,     2,      2,  ...   1,  1,   4,    5,     6,      7,  ...   1,  1,  10,   23,    46,     82,  ...   1,  1,  36,  239,  1086,   3707,  ...   1,  1, 202, 5828, 79326, 642457,  ... MAPLE b:= proc(n, j, k) local nn;       nn:= n+1;       if n<0  then 0     elif j=0  or n=0 or k<=1 then 1     elif j=1  then nn     elif n>=j then (nn-j) *binomial(nn, j) *add(binomial(j, h)                    /(nn-j+h) *b(j-h-1, j, k) *(-1)^h, h=0..j-1)               else b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)       fi     end: A:= (n, k)-> b(1, n, k): seq(seq(A(n, d-n), n=0..d), d=0..13); MATHEMATICA b[n_, j_, k_] := Module[{nn = n+1}, Which[n < 0, 0, j == 0 || n == 0 || k <= 1, 1, j == 1, nn, n >= j, (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)* b[j-h-1, j, k]*(-1)^h, {h, 0, j-1}], True, b[n, j, k] = b[n-1, j, k] + b[k*n, j-1, k] ] ]; a[n_, k_] := b[1, n, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *) CROSSREFS Columns k=0+1, 2-10 give: A000012, A002577, A078125, A078537, A111822, A111827, A111832, A111837, A145512, A145513. Row n=3 gives: A189890(k+1). Main diagonal gives: A145514. Cf. A007318. Sequence in context: A295679 A287214 A287216 * A267383 A272896 A188919 Adjacent sequences:  A145512 A145513 A145514 * A145516 A145517 A145518 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Oct 11 2008 EXTENSIONS Edited by Alois P. Heinz, Jan 12 2011 STATUS approved

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Last modified October 15 22:25 EDT 2019. Contains 328038 sequences. (Running on oeis4.)