A323574 G.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * ((1+x)^n - 1)^n / (A(x) + 2 - 2*(1+x)^n)^(n+1).

1, 2, 20, 320, 7996, 269272, 11293792, 563253696, 32433009160, 2113347523336, 153579286783456, 12309659862402976, 1078628781953636960, 102578628758305245024, 10523148808846566898816, 1158407291029244188955264, 136214299772837816557703120, 17040721610970237566148646464, 2260018461602151565512432884608, 316748455363386162460484685488512

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} 2^n * ((1+x)^n - 1)^n/(A(x) + 2 - 2*(1+x)^n)^(n+1).

(2) 1 = Sum_{n>=0} 2^n * ((1+x)^n + 1)^n/(A(x) + 2 + 2*(1+x)^n)^(n+1).

Example

G.f.: A(x) = 1 + 2*x + 20*x^2 + 320*x^3 + 7996*x^4 + 269272*x^5 + 11293792*x^6 + 563253696*x^7 + 32433009160*x^8 + 2113347523336*x^9 + 153579286783456*x^10 + ...
such that
1 = 1/A(x) + 2*((1+x) - 1)/(A(x) + 2 - 2*(1+x))^2  +  2^2*((1+x)^2 - 1)^2/(A(x) + 2 - 2*(1+x)^2)^3  +  2^3*((1+x)^3 - 1)^3/(A(x) + 2 - 2*(1+x)^3)^4  +  2^4*((1+x)^4 - 1)^4/(A(x) + 2 - 2*(1+x)^4)^5  +  2^5*((1+x)^5 - 1)^5/(A(x) + 2 - 2*(1+x)^5)^6 + ...
also,
1 = 1/(A(x) + 4)  +  2*(1 + (1+x))/(A(x) + 2 + 2*(1+x))^2  +  2^2*(1 + (1+x)^2)^2/(A(x) + 2 + 2*(1+x)^2)^3  +  2^3*(1 + (1+x)^3)^3/(A(x) + 2 + 2*(1+x)^3)^4  +  2^4*(1 + (1+x)^4)^4/(A(x) + 2 + 2*(1+x)^4)^5  +  2^5*(1 + (1+x)^5)^5/(A(x) + 2 + 2*(1+x)^5)^6 + ...

Prog

(PARI) {a(n) = my(A=[1], X=x+x*O(x^n)); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 2^m * ((1+X)^m - 1)^m / (Ser(A) + 2 - 2*(1+X)^m)^(m+1) ) )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A323313.

Sequence in context: A367862 A177397 A360342 * A294454 A322729 A304861

Adjacent sequences: A323571 A323572 A323573 * A323575 A323576 A323577

Keyword

nonn

Author

Paul D. Hanna, Feb 06 2019

Status

approved