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.

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

T. D. Noe, Table of n, a(n) for n=0..35

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).

Cf. A002449

Sequence in context: A361057 A013108 A173945 * A290000 A363423 A102922

Adjacent sequences: A111843 A111844 A111845 * A111847 A111848 A111849

Keyword

nonn

Author

Paul D. Hanna, Aug 23 2005

Status

approved