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%I #11 Jun 13 2017 22:34:19
%S 1,5,55,2055,291430,165397680,390075741430,3927972221522680,
%T 172358768282285194555,33479766506261422878944555,
%U 29150234311482124092454001991430
%N 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.
%C 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)).
%H Alois P. Heinz, <a href="/A111821/b111821.txt">Table of n, a(n) for n = 0..40</a>
%F a(n) = [x^(4*5^n)] 1/Product_{j>=0}(1-x^(5^j)).
%o (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]))
%Y Cf. A111820 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111826 (q=6), A111831 (q=7), A111836 (q=8).
%K nonn
%O 0,2
%A _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005
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