A111821 Number of partitions of 4*5^n into powers of 5, also equals column 1 of triangle A111820, which shifts columns left and up under matrix 5th power.

1, 5, 55, 2055, 291430, 165397680, 390075741430, 3927972221522680, 172358768282285194555, 33479766506261422878944555, 29150234311482124092454001991430

Offset: 0

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

• nonn

Comments

Let q=5; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..40

Crossrefs

Cf. A111820 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111826 (q=6), A111831 (q=7), A111836 (q=8).

Programs

PARI

a(n,q=5)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(A[n+2,2]))

Formula

a(n) = [x^(4*5^n)] 1/Product_{j>=0}(1-x^(5^j)).