|
| |
|
|
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.
|
|
18
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 0,8
|
|
|
LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..860
|
|
|
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);
|
|
|
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: A129176 A134132 A030424 * A188919 A026519 A025177
Adjacent sequences: A145512 A145513 A145514 * A145516 A145517 A145518
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 11 2008
|
|
|
EXTENSIONS
| Edited by Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jan 12 2011
|
| |
|
|
