A182135 -id:A182135 - cp's OEIS frontend

A172288 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^2^n into powers of 2 less than or equal to 2^k.

1, 2, 1, 2, 3, 1, 2, 4, 9, 1, 2, 4, 25, 129, 1, 2, 4, 35, 4225, 32769, 1, 2, 4, 36, 47905, 268468225, 2147483649, 1, 2, 4, 36, 222241, 733276217345, 1152921506754330625, 9223372036854775809, 1

Offset: 0

Alois P. Heinz, Jan 26 2011

A(18,18) = 2797884726...4715787265 has 1420371 decimal digits and was computed by the algorithm given below.

			A(2,1) = 9, because there are 9 partitions of 2^2^2=16 into powers of 2 less than or equal to 2^1=2: [2,2,2,2,2,2,2,2], [2,2,2,2,2,2,2,1,1], [2,2,2,2,2,2,1,1,1,1], [2,2,2,2,2,1,1,1,1,1,1], [2,2,2,2,1,1,1,1,1,1,1,1], [2,2,2,1,1,1,1,1,1,1,1,1,1], [2,2,1,1,1,1,1,1,1,1,1,1,1,1], [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1,     2,         2,            2,               2,  ...
  1,     3,         4,            4,               4,  ...
  1,     9,        25,           35,              36,  ...
  1,   129,      4225,        47905,          222241,  ...
  1, 32769, 268468225, 733276217345, 751333186150401,  ...

Alois P. Heinz, Table of n, a(n) for n = 0..77

Cf. A002577, A000123, A181322, A145515.

Main diagonal gives: A182135.

b:= proc(n,j) option remember; local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(2^(2^n-k), k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, j_] := 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_] := b[2^(2^n-k), k]; Table[Table[a[n, d-n] // FullSimplify, {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

A(n,k) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..k-1} (1-x^(2^j)).

A212775 Number of partitions of 2^(2^n) into powers of 2.

Original entry on oeis.org

2, 4, 36, 692004, 114788185359199234852802340

Offset: 0

Alois P. Heinz, May 26 2012

nonn

Lengths (in decimal digits) of the terms a(0), a(1), ... are: 1, 1, 2, 6, 27, 119, 525, 2241, 9330, ... .

			a(0) = 2 because the number of partitions of 2^2^0 = 2 into powers of 2 is 2: [2], [1,1].
a(1) = 4: [4], [2,2], [2,1,1], [1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..6

Cf. A000123, A002577, A172288, A182135.

b:= proc(n, j) option remember; local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(1, 2^n):
seq(a(n), n=0..6);

b[n_, j_] := 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_] := b[1, 2^n]; Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Feb 28 2017, translated from Maple *)

a(n) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..2^n-1} (1-x^(2^j)).

cp's OEIS Frontend

A172288 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^2^n into powers of 2 less than or equal to 2^k.

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