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A145515
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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.
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22
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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
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OFFSET
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0,8
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LINKS
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FORMULA
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See program.
For k>1: A(n,k) = [x^(k^n)] 1/Product_{j>=0} (1-x^(k^j)).
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0+1, 2-10 give: A000012, A002577, A078125, A078537, A111822, A111827, A111832, A111837, A145512, A145513.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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