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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.

%I A152977 #45 Oct 04 2018 19:50:46
%S A152977 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,
%T A152977 35,81,33,1,1,2,4,10,36,165,289,65,1,1,2,4,10,36,201,969,1089,129,1,1,
%U A152977 2,4,10,36,202,1625,6545,4225,257,1,1,2,4,10,36,202,1827,17361,47905,16641,513,1
%N 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.
%C A152977 Column sequences converge towards A002577.
%H A152977 Alois P. Heinz, <a href="/A152977/b152977.txt">Antidiagonals n = 0..140, flattened</a>
%F A152977 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.
%e A152977 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].
%e A152977 Square array A(n,k) begins:
%e A152977   1,  1,  1,   1,   1,   1,  ...
%e A152977   1,  2,  2,   2,   2,   2,  ...
%e A152977   1,  3,  4,   4,   4,   4,  ...
%e A152977   1,  5,  9,  10,  10,  10,  ...
%e A152977   1,  9, 25,  35,  36,  36,  ...
%e A152977   1, 17, 81, 165, 201, 202,  ...
%p A152977 b:= proc(n,j) local nn, r;
%p A152977       if n<0 then 0
%p A152977     elif j=0 then 1
%p A152977     elif j=1 then n+1
%p A152977     elif n<j then b(n,j):= b(n-1, j) +b(2*n, j-1)
%p A152977              else nn:= 1 +floor(n);
%p A152977                   r:= n-nn;
%p A152977                   (nn-j) *binomial(nn, j) *add(binomial(j, h)
%p A152977                   /(nn-j+h) *b(j-h+r, j) *(-1)^h, h=0..j-1)
%p A152977       fi
%p A152977     end:
%p A152977 A:= (n,k)-> `if`(n=0, 1, b(2^(n-k), k)):
%p A152977 seq(seq(A(n, d-n), n=0..d), d=0..11);
%t A152977 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 *)
%Y A152977 Columns k=0-10 give: A000012, A094373, A028400(n-2) for n>1, A210772, A210773, A210774, A210775, A210776, A210777, A210778, A210779.
%Y A152977 Cf. A002577, A000123, A181322, A145515.
%Y A152977 Main diagonal and lower diagonals give: A002577, A125792, A125794.
%K A152977 nonn,tabl
%O A152977 0,5
%A A152977 _Alois P. Heinz_, Jan 26 2011