A136639 G.f. A(x) satisfies: 1 + 4x = Sum_{n>=0} x^n * [y^n] A(y)^{2^n}, so that the coefficient of y^n in the (2^n)-th power of A(y) is zero for n>1.

1, 2, -6, 28, -410, -40548, -25678044, -46695237064, -261442466307066, -4697327493273424596, -282093987534237286507572, -58573002543105566470447978872, -43150254003464729476289222051517988, -114953354885565948577897809039842112360424

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Formula

G.f. A(x) satisfies: 1 + 4x = Sum_{n>=0} log( A(2^n*x) )^n / n!.

Example

A(x) = 1 + 2*x - 6*x^2 + 28*x^3 - 410*x^4 - 40548*x^5 - 25678044*x^6 -...
Illustrate 1 + 4x = Sum_{n>=0} x^n * [y^n] A(y)^{2^n} by listing
the initial coefficients in the (2^n)-th powers of A(x):
A^(2^0)=[1, 2, -6, 28, -410, -40548, -25678044, -46695237064,...];
A^(2^1)=[1, 4, -8, 32, -672, -83072, -51512576, -93492722688,...];
A^(2^2)=[1, 8, 0, 0, -1024, -172032, -103677952, -187396259840,...];
A^(2^3)=[1, 16, 64, 0, -2048, -360448, -210108416, -376451366912,...];
A^(2^4)=[1, 32, 384, 2048, 0, -786432, -432013312, -759672340480,...];
A^(2^5)=[1, 64, 1792, 28672, 278528, 0, -910163968, -1547597512704,...];
A^(2^6)=[1, 128, 7680, 286720, 7438336, 138412032, 0, -3195724103680,.];
A^(2^7)=[1, 256, 31744, 2539520, 147259392, 6585057280, 231894679552, 0,.];
the main diagonal is all zeros except for the initial [1,4].

Prog

(PARI) {a(n)=local(A=[1]); if(n==1, 2, for(i=0, n, A=Vec(Ser(concat(Vec(Ser(A)^(2^#A)), if(#A==1, 1, 0)))^(1/2^#A))); A[n+1]*4^n)}

Crossrefs

Sequence in context: A325507 A306793 A326359 * A368003 A368143 A284594

Adjacent sequences: A136636 A136637 A136638 * A136640 A136641 A136642

Keyword

sign

Author

Paul D. Hanna, Jan 19 2008

Status

approved