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 A181322 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. 10
 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 2, 4, 6, 5, 1, 1, 2, 4, 6, 9, 6, 1, 1, 2, 4, 6, 10, 12, 7, 1, 1, 2, 4, 6, 10, 14, 16, 8, 1, 1, 2, 4, 6, 10, 14, 20, 20, 9, 1, 1, 2, 4, 6, 10, 14, 20, 26, 25, 10, 1, 1, 2, 4, 6, 10, 14, 20, 26, 35, 30, 11, 1, 1, 2, 4, 6, 10, 14, 20, 26, 36, 44, 36, 12, 1, 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 56, 42, 13, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Column sequences converge towards A000123. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 4. FORMULA G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(2^j)). A(n,k) = Sum_{i=0..k} A089177(n,i). For n < 2^k, T(n,k) = A000123(k). T(n,0) = 1, T(n,1) = n+1. - M. F. Hasler, Feb 19 2019 EXAMPLE A(3,2) = 6, because there are 6 partitions of 2*3=6 into powers of 2 less than or equal to 2^2=4: [4,2], [4,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1], [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, 4, 6, 6, 6, 6, ... 1, 5, 9, 10, 10, 10, ... 1, 6, 12, 14, 14, 14, ... MAPLE 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 b(n/2^(k-1), k): seq(seq(A(n, d-n), n=0..d), d=0..13); MATHEMATICA b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n<0, 0, j == 0, 1, j == 1, n+1, n

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Last modified January 31 18:34 EST 2023. Contains 359980 sequences. (Running on oeis4.)