login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 20
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

FORMULA

See program.

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

EXAMPLE

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,  ...

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A134132 A030424 A216656 * A267383 A272896 A188919

Adjacent sequences:  A145512 A145513 A145514 * A145516 A145517 A145518

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Oct 11 2008

EXTENSIONS

Edited by Alois P. Heinz, Jan 12 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified April 27 10:54 EDT 2017. Contains 285512 sequences.