A262553 -id:A262553 - cp's OEIS frontend

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

Alois P. Heinz, Jan 26 2011

Column sequences converge towards A000123.

			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,  ...

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.

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).

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

b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n<0, 0, j == 0, 1, j == 1, n+1, nJean-François Alcover, Jan 15 2014, translated from Maple *)

A181322(n,k,r=1)={if(nA181322(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

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

A262554 Triangle read by rows: coefficients in Bell's formula for number of ways of making change when coins have denominations which are powers of 2.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 9, 16, 8, 1, 35, 130, 160, 64, 1, 201, 1424, 3272, 3072, 1024, 1, 1827, 23682, 91040, 151104, 114688, 32768, 1, 27337, 637328, 3872456, 10136576, 13165568, 8388608, 2097152, 1, 692003, 28867714, 268176288, 1041587776, 2061746176

Offset: 1

N. J. A. Sloane, Oct 09 2015

			Triangle begins:
  1;
  1,    1;
  1,    3,     2;
  1,    9,    16,     8;
  1,   35,   130,   160,     64;
  1,  201,  1424,  3272,   3072,   1024;
  1, 1827, 23682, 91040, 151104, 114688, 32768;
  ...

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

A002575 and A002576 are diagonals.

Cf. A181322, A262553.

More terms from Sean A. Irvine, Oct 19 2015

cp's OEIS Frontend

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.

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