A145514 -id:A145514 - cp's OEIS frontend

A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 10, 1, 1, 1, 2, 6, 23, 36, 1, 1, 1, 2, 7, 46, 239, 202, 1, 1, 1, 2, 8, 82, 1086, 5828, 1828, 1, 1, 1, 2, 9, 134, 3707, 79326, 342383, 27338, 1, 1, 1, 2, 10, 205, 10340, 642457, 18583582, 50110484, 692004, 1, 1, 1, 2, 11, 298, 24901, 3649346, 446020582, 14481808030, 18757984046, 30251722, 1, 1

Offset: 0

Alois P. Heinz, Oct 11 2008

			A(2,3) = 5, because there are 5 partitions of 3^2=9 into powers of 3: [1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,3], [1,1,1,3,3], [3,3,3], [9].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,  ...
  1,  1,   2,    2,     2,      2,  ...
  1,  1,   4,    5,     6,      7,  ...
  1,  1,  10,   23,    46,     82,  ...
  1,  1,  36,  239,  1086,   3707,  ...
  1,  1, 202, 5828, 79326, 642457,  ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened

Columns k=0+1, 2-10 give: A000012, A002577, A078125, A078537, A111822, A111827, A111832, A111837, A145512, A145513.
Row n=3 gives: A189890(k+1).
Main diagonal gives: A145514.
Cf. A007318.

b:= proc(n, j, k) local nn;
      nn:= n+1;
      if n<0  then 0
    elif j=0  or n=0 or k<=1 then 1
    elif j=1  then nn
    elif n>=j then (nn-j) *binomial(nn, j) *add(binomial(j, h)
                   /(nn-j+h) *b(j-h-1, j, k) *(-1)^h, h=0..j-1)
              else b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)
      fi
    end:
A:= (n, k)-> b(1, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..13);

b[n_, j_, k_] := Module[{nn = n+1}, Which[n < 0, 0, j == 0 || n == 0 || k <= 1, 1, j == 1, nn, n >= j, (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)* b[j-h-1, j, k]*(-1)^h, {h, 0, j-1}], True, b[n, j, k] = b[n-1, j, k] + b[k*n, j-1, k] ] ]; a[n_, k_] := b[1, n, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

See program.

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

Edited by Alois P. Heinz, Jan 12 2011

A196879 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of n^k into powers of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 10, 1, 1, 1, 1, 6, 23, 36, 1, 1, 1, 1, 9, 72, 132, 94, 1, 1, 1, 1, 16, 335, 1086, 729, 284, 1, 1, 1, 1, 36, 2220, 15265, 15076, 3987, 692, 1, 1, 1, 1, 85, 19166, 374160, 642457, 182832, 18687, 1828, 1, 1

Offset: 0

Alois P. Heinz, Oct 07 2011

			A(2,3) = 3, because the number of partitions of 2^3=8 into powers of 3 is 3: [1,1,3,3], [1,1,1,1,1,3], [1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,   1,     1,      1,  ...
  1,  1,  1,   1,     1,      1,  ...
  1,  1,  4,   3,     6,      9,  ...
  1,  1, 10,  23,    72,    335,  ...
  1,  1, 36, 132,  1086,  15265,  ...
  1,  1, 94, 729, 15076, 642457,  ...

Alois P. Heinz, Antidiagonals n = 0..44, flattened

Columns k=0+1, 2-10 give: A000012, A196880, A196881, A196882, A196883, A196884, A196885, A196886, A196887, A196888.
Rows n=0+1, 2-10 give: A000012, A196889, A196890, A196891, A196892, A196893, A196894, A196895, A196896, A196897.
Main diagonal gives: A145514.
Cf. A145515.

b:= proc(n, j, k) local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n

a[, 0] = a[, 1] = a[0, ] = a[1, ] = 1; a[n_, k_] := SeriesCoefficient[ 1/Product[ (1 - x^(k^j)), {j, 0, n}], {x, 0, n^k}]; Table[a[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 09 2013 *)

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

A331402 a(n) = [x^(n^n)] Product_{k>=1} 1 / (1 - x^(k^n)).

Original entry on oeis.org

1, 2, 5, 36, 1104, 140549, 82159688, 237700614212, 3591644060379486

Offset: 1

Ilya Gutkovskiy, Jan 16 2020

Number of partitions of n^n into n-th powers.

			a(2) = 2 because we have [4] and [1, 1, 1, 1].

Diagonal of A259799.

Cf. A145514.

a(n) = A259799(n,n).

a(8)-a(9) from Giovanni Resta, Jan 17 2020

A337990 Number of compositions (ordered partitions) of n^n into powers of n.

Original entry on oeis.org

1, 1, 6, 26426, 773527571233557154337704151068262296

Offset: 0

Ilya Gutkovskiy, Oct 06 2020

nonn

The next term is too large to include.

			a(2) = 6 because 2^2 = 4 and we have [4], [2, 2], [2, 1, 1] (3 permutations), [1, 1, 1, 1] and 1 + 1 + 3 + 1 = 6.

Index entries for sequences related to compositions

Cf. A145514, A248377.

Join[{1, 1}, Table[SeriesCoefficient[1/(1 - Sum[x^(n^k), {k, 0, n}]), {x, 0, n^n}], {n, 2, 4}]]

a(n) = [x^(n^n)] 1 / (1 - Sum_{k>=0} x^(n^k)), for n > 1.

cp's OEIS Frontend

A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

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A196879 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of n^k into powers of k.

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A331402 a(n) = [x^(n^n)] Product_{k>=1} 1 / (1 - x^(k^n)).

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A337990 Number of compositions (ordered partitions) of n^n into powers of n.

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