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A073266
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Triangle read by rows: T(n,k) is the number of compositions of n as the sum of k integral powers of 2.
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5
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1, 1, 1, 0, 2, 1, 1, 1, 3, 1, 0, 2, 3, 4, 1, 0, 2, 4, 6, 5, 1, 0, 0, 6, 8, 10, 6, 1, 1, 1, 3, 13, 15, 15, 7, 1, 0, 2, 3, 12, 25, 26, 21, 8, 1, 0, 2, 6, 10, 31, 45, 42, 28, 9, 1, 0, 0, 6, 16, 30, 66, 77, 64, 36, 10, 1, 0, 2, 4, 18, 40, 76, 126, 126, 93, 45, 11, 1, 0, 0, 6, 16, 50, 96, 168, 224, 198, 130, 55, 12, 1
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OFFSET
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1,5
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COMMENTS
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LINKS
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FORMULA
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T(n, k) = coefficient of x^n in the formal power series (x + x^2 + x^4 + x^8 + x^16 + ...)^k. - Emeric Deutsch, Feb 04 2005
T(0, k) = T(n, 0) = 0, T(n, k) = 0 if k > n, T(n, 1) = 1 if n = 2^m, 0 otherwise and in other cases T(n, k) = Sum_{i=0..floor(log_2(n-1))} T(n-(2^i), k-1). - Emeric Deutsch, Feb 04 2005
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EXAMPLE
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T(6,3) = 4 because there are four ordered partitions of 6 into 3 powers of 2, namely: 4+1+1, 1+4+1, 1+1+4 and 2+2+2.
Triangle begins:
1;
1, 1;
0, 2, 1;
1, 1, 3, 1;
0, 2, 3, 4, 1;
0, 2, 4, 6, 5, 1;
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MAPLE
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b:= proc(n) option remember; expand(`if`(n=0, 1,
add(b(n-2^j)*x, j=0..ilog2(n))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)):
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> if n = 2^ilog2(n) then 1 else 0 fi); # Peter Luschny, Oct 07 2022
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MATHEMATICA
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m:= 10; T[n_, k_]:= T[n, k]= Coefficient[(Sum[x^(2^j), {j, 0, m+1}])^k, x, n]; Table[T[n, k], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Mar 06 2020 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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