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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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,  ...

MAPLE

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<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)-> b(2^(2^n-k), k):

seq(seq(A(n, d-n), n=0..d), d=0..8);

MATHEMATICA

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-Fran├žois Alcover, Dec 11 2013, translated from Maple *)

CROSSREFS

Cf. A002577, A000123, A181322, A145515.

Main diagonal gives: A182135.

Sequence in context: A286880 A178030 A131879 * A134628 A064882 A065158

Adjacent sequences:  A172285 A172286 A172287 * A172289 A172290 A172291

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jan 26 2011

STATUS

approved

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Last modified March 29 21:32 EDT 2020. Contains 333117 sequences. (Running on oeis4.)