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A152977
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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 2^n into powers of 2 less than or equal to 2^k.
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13
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 9, 9, 1, 1, 2, 4, 10, 25, 17, 1, 1, 2, 4, 10, 35, 81, 33, 1, 1, 2, 4, 10, 36, 165, 289, 65, 1, 1, 2, 4, 10, 36, 201, 969, 1089, 129, 1, 1, 2, 4, 10, 36, 202, 1625, 6545, 4225, 257, 1, 1, 2, 4, 10, 36, 202, 1827, 17361, 47905, 16641, 513, 1
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OFFSET
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0,5
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COMMENTS
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Column sequences converge towards A002577.
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LINKS
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FORMULA
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A(n,k) = [x^2^(n-1)] 1/(1-x) * 1/Product_{j=0..k-1} (1-x^(2^j)) for n>0; A(0,k) = 1.
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EXAMPLE
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A(3,2) = 9, because there are 9 partitions of 2^3=8 into powers of 2 less than or equal to 2^2=4: [4,4], [4,2,2], [4,2,1,1], [4,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, ...
1, 3, 4, 4, 4, 4, ...
1, 5, 9, 10, 10, 10, ...
1, 9, 25, 35, 36, 36, ...
1, 17, 81, 165, 201, 202, ...
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MAPLE
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b:= proc(n, j) local nn, r;
if n<0 then 0
elif j=0 then 1
elif j=1 then n+1
elif n<j then b(n, j):= b(n-1, j) +b(2*n, j-1)
else nn:= 1 +floor(n);
r:= n-nn;
(nn-j) *binomial(nn, j) *add(binomial(j, h)
/(nn-j+h) *b(j-h+r, j) *(-1)^h, h=0..j-1)
fi
end:
A:= (n, k)-> `if`(n=0, 1, b(2^(n-k), k)):
seq(seq(A(n, d-n), n=0..d), d=0..11);
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MATHEMATICA
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b[n_, j_] := Module[{nn, r}, Which[n < 0, 0, j == 0, 1, j == 1, n+1, n < j, b[n, j] = b[n-1, j]+b[2*n, j-1], True, nn = 1+Floor[n]; r := n-nn; (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)*b[j-h+r, j]*(-1)^h, {h, 0, j-1}]]]; a[n_, k_] := If[n == 0, 1, b[2^(n-k), k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000012, A094373, A028400(n-2) for n>1, A210772, A210773, A210774, A210775, A210776, A210777, A210778, A210779.
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KEYWORD
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AUTHOR
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STATUS
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approved
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