A152977 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.

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

Offset: 0

Author

Alois P. Heinz, Jan 26 2011

Keywords

• nonn
• tabl

Comments

Column sequences converge towards A002577.

Examples

			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,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000012, A094373, A028400(n-2) for n>1, A210772, A210773, A210774, A210775, A210776, A210777, A210778, A210779.

Cf. A002577, A000123, A181322, A145515.

Main diagonal and lower diagonals give: A002577, A125792, A125794.

Programs

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 `if`(n=0, 1, b(2^(n-k), k)):
seq(seq(A(n, d-n), n=0..d), d=0..11);

Mathematica

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 *)

Formula

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.