A181322 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, 4, 1, 1, 2, 4, 6, 5, 1, 1, 2, 4, 6, 9, 6, 1, 1, 2, 4, 6, 10, 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, 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

Offset

0,5

Comments

Column sequences converge towards A000123.

Links

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

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 4.

Formula

G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(2^j)).

A(n,k) = Sum_{i=0..k} A089177(n,i).

For n < 2^k, T(n,k) = A000123(k). T(n,0) = 1, T(n,1) = n+1. - M. F. Hasler, Feb 19 2019

Example

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].
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,  4,  6,  6,  6,  6,  ...
  1,  5,  9, 10, 10, 10,  ...
  1,  6, 12, 14, 14, 14,  ...

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<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(n/2^(k-1), k):
seq(seq(A(n, d-n), n=0..d), d=0..13);

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 = 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 *)

Prog

(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

Crossrefs

Columns k=0-5 give: A000012, A000027(n+1), A002620(n+2), A008804, A088932, A088954.

Main diagonal gives A000123.

Cf. A145515.

See A262553 for another version of this array.

See A072170 for a related array (having the same limiting column).

Sequence in context: A140207 A104763 A027751 * A004070 A180562 A199711

Adjacent sequences: A181319 A181320 A181321 * A181323 A181324 A181325

Keyword

nonn,tabl

Author

Alois P. Heinz, Jan 26 2011

Status

approved