A303288 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 5*x*A(x) )^n * 2^n / 3^(n+1).

1, 14, 688, 56738, 6347176, 881241656, 144796770004, 27351977086556, 5826096152426212, 1380051673281134312, 359720002818554238352, 102317793242070983628176, 31540355035889303797419616, 10475792506313141986771902704, 3730248479020018845292570520560, 1417811189172027111629537752756520, 572992474515466430293335350543824096

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..50

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} 2^n * ( (1+x)^n - 5*x*A(x) )^n / 3^(n+1).

(2) 1 = Sum_{n>=0} 2^n * (1+x)^(n^2) / (3 + 10*x*A(x)*(1+x)^n)^(n+1). - Paul D. Hanna, Jan 10 2019

(3) 1 = Sum_{k>=0} (-5*x)^k * A(x)^k * Sum_{n>=0} C(n+k,k) * (1+x)^(n*(n+k)) * 2^(n+k) / 3^(n+k+1).

(4) 1 = Sum_{n>=0} Sum_{k=0..n} C(n,k) * (1+x)^(n*(n-k)) * 2^n / 3^(n+1) * (-5*x)^k * A(x)^k.

Example

G.f.: A(x) = 1 + 14*x + 688*x^2 + 56738*x^3 + 6347176*x^4 + 881241656*x^5 + 144796770004*x^6 + 27351977086556*x^7 + 5826096152426212*x^8 + ...
such that
1 = 1/3 + 2*((1+x) - 5*x*A(x))/3^2 + 2^2*((1+x)^2 - 5*x*A(x))^2/3^3 + 2^3*((1+x)^3 - 5*x*A(x))^3/3^4 + 2^4*((1+x)^4 - 5*x*A(x))^4/3^5 + 2^5*((1+x)^5 - 5*x*A(x))^5/3^6 + ...
also,
1 = 1/(3 + 10*x*A(x)) + 2*(1+x)/(3 + 10*x*A(x))^2 + 2^2*(1+x)^4/(3 + 10*x*A(x))^3 + 2^3*(1+x)^9/(3 + 10*x*A(x))^4 + 2^4*(1+x)^16/(3 + 10*x*A(x))^5 + 2^5*(1+x)^25/(3 + 10*x*A(x))^6 + ...

Crossrefs

Cf. A301435, A303291, A323314, A323315, A323316, A323317, A323318, A323319, A323320, A323321.

Sequence in context: A336513 A233078 A223129 * A277299 A233506 A103426

Adjacent sequences: A303285 A303286 A303287 * A303289 A303290 A303291

Keyword

nonn

Author

Paul D. Hanna, Apr 23 2018

Status

approved