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A111846 Number of partitions of 4^n - 1 into powers of 4, also equals column 0 of triangle A111845, which shifts columns left and up under matrix 4th power.

%I A111846 #12 Jul 11 2025 14:05:00
%S A111846 1,1,4,40,1040,78240,18504256,14463224448,38544653734144,
%T A111846 357896006503348736,11766320092785122862080,
%U A111846 1387031702368547767793690624,592262859312707222259571097997312
%N A111846 Number of partitions of 4^n - 1 into powers of 4, also equals column 0 of triangle A111845, which shifts columns left and up under matrix 4th power.
%C A111846 a(n) equals the partitions of 4^n-1 into powers of 4, or, the coefficient of x^(4^n-1) in 1/Product_{j>=0}(1-x^(4^j)).
%H A111846 T. D. Noe, <a href="/A111846/b111846.txt">Table of n, a(n) for n=0..35</a>
%F A111846 G.f.: A(x) = 1 + Sum_{n>=1} (1/n!)*Product_{j=0..n-1} L(4^j*x) where L(x) satisfies: x = Sum_{n>=1} -(-1)^n/n!*Product_{j=0..n-1} L(4^j*x); L(x) equals the g.f. of column 0 of the matrix log of P (A111849).
%e A111846 G.f. A(x) = 1 + L(x) + L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3!
%e A111846 + L(x)*L(4*x)*L(4^2*x)*L(4^3*x)/4! + ...
%e A111846 where L(x) satisfies:
%e A111846 x = L(x) - L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3! +- ...
%e A111846 and L(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4 +....(A111849).
%o A111846 (PARI) {a(n,q=4)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,j]=(A^q)[i-1,1], B[i,j]=(A^q)[i-1,j-1]));));A=B);return(A[n+1,1]))}
%Y A111846 Cf. A111845 (triangle).
%Y A111846 Cf. A002449
%K A111846 nonn
%O A111846 0,3
%A A111846 _Paul D. Hanna_, Aug 23 2005