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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.
3
1, 1, 4, 40, 1040, 78240, 18504256, 14463224448, 38544653734144, 357896006503348736, 11766320092785122862080, 1387031702368547767793690624, 592262859312707222259571097997312
OFFSET
0,3
COMMENTS
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)).
LINKS
FORMULA
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).
EXAMPLE
G.f. A(x) = 1 + L(x) + L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3!
+ L(x)*L(4*x)*L(4^2*x)*L(4^3*x)/4! + ...
where L(x) satisfies:
x = L(x) - L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3! -+ ...
and L(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4 +....(A111849).
PROG
(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]))}
CROSSREFS
Cf. A111845 (triangle).
Sequence in context: A361057 A013108 A173945 * A290000 A363423 A102922
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 23 2005
STATUS
approved