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%I #42 Feb 19 2019 13:02:28
%S 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,
%T 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,
%U 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
%N 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 Column sequences converge towards A000123.
%H Alois P. Heinz, <a href="/A181322/b181322.txt">Antidiagonals n = 0..140, flattened</a>
%H G. Blom and C.-E. Froeberg, <a href="/A002575/a002575.pdf">Om myntvaexling (On money-changing) [Swedish]</a>, Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 4.
%F G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(2^j)).
%F A(n,k) = Sum_{i=0..k} A089177(n,i).
%F For n < 2^k, T(n,k) = A000123(k). T(n,0) = 1, T(n,1) = n+1. - _M. F. Hasler_, Feb 19 2019
%e 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].
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 2, 2, 2, 2, ...
%e 1, 3, 4, 4, 4, 4, ...
%e 1, 4, 6, 6, 6, 6, ...
%e 1, 5, 9, 10, 10, 10, ...
%e 1, 6, 12, 14, 14, 14, ...
%p b:= proc(n, j) local nn, r;
%p if n<0 then 0
%p elif j=0 then 1
%p elif j=1 then n+1
%p elif n<j then b(n, j):= b(n-1, j) +b(2*n, j-1)
%p else nn:= 1 +floor(n);
%p r:= n-nn;
%p (nn-j) *binomial(nn, j) *add(binomial(j, h)
%p /(nn-j+h) *b(j-h+r, j) *(-1)^h, h=0..j-1)
%p fi
%p end:
%p A:= (n, k)-> b(n/2^(k-1), k):
%p seq(seq(A(n, d-n), n=0..d), d=0..13);
%t 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 = Floor[n]+1; r = n - nn; (nn-j)*Binomial[nn, j]*Sum[(Binomial[j, h]*b[j-h+r, j]*(-1)^h)/(nn-j+h), {h, 0, j-1}]]]; A[n_, k_] := b[n/2^(k-1), k]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* _Jean-François Alcover_, Jan 15 2014, translated from Maple *)
%o (PARI) A181322(n,k,r=1)={if(n<r,r,!k,1, r&&n/=2^(k-1); k==1, n+1, n<k, A181322(n-1,k,0)+A181322(2*n,k-1,0),n-=r=1+n\1,(r-k)*binomial(r,k)*sum(i=0,min(k-1,k+n), binomial(k,i)/(r-k+i)*A181322(k-i+n,k,0) *(-1)^i))} \\ From Maple. - _M. F. Hasler_, Feb 19 2019
%Y Columns k=0-5 give: A000012, A000027(n+1), A002620(n+2), A008804, A088932, A088954.
%Y Main diagonal gives A000123.
%Y Cf. A145515.
%Y See A262553 for another version of this array.
%Y See A072170 for a related array (having the same limiting column).
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Jan 26 2011
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