A301307 - OEIS

%I #14 Mar 24 2018 16:32:49

%S 1,5,98,3239,150176,8958473,653364947,56325265925,5603297711741,

%T 631787569243643,79620187792726844,11090608163844996365,

%U 1692024644610151317068,280593919265423518611017,50255068227934275890880470,9667645123441963396364779439,1988058929295585346059732920903,435204469378969786061222253686549,101044871217450582545711556498557285

%N G.f.: Sum_{n>=0} (1 + (1+x)^n)^n / 3^(n+1).

%H Paul D. Hanna, <a href="/A301307/b301307.txt">Table of n, a(n) for n = 0..195</a>

%F G.f.: Sum_{n>=0} (1+x)^(n^2) / (3 - (1+x)^n)^(n+1).

%F G.f.: Sum_{n>=0} Sum_{k=0..n} binomial(n,k) * (1 + x)^(n*k) / 3^(n+1).

%F a(n) = Sum_{j>=0} Sum_{k=0..j} binomial(j, k) * binomial(j*k, n) / 3^(j+1).

%F a(n) ~ c * d^n * n^n, where d = 4.88100884940898277361223446294548499145552953621086588549015342712172151... and c = 1.0401387348267211789387929284813380774183533880659572052994951... - _Vaclav Kotesovec_, Mar 22 2018

%e G.f.: A(x) = 1 + 5*x + 98*x^2 + 3239*x^3 + 150176*x^4 + 8958473*x^5 + 653364947*x^6 + 56325265925*x^7 + 5603297711741*x^8 + ...

%e such that

%e A(x) = 1/3 + (1 + (1+x))/3^2 + (1 + (1+x)^2)^2/3^3 + (1 + (1+x)^3)^3/3^4 + (1 + (1+x)^4)^4/3^5 + (1 + (1+x)^5)^5/3^6 + (1 + (1+x)^6)^6/3^7 + ...

%e Also,

%e A(x) = 1/2 + (1+x)/(3 - (1+x))^2 + (1+x)^4/(3 - (1+x)^2)^3 + (1+x)^9/(3 - (1+x)^3)^4 + (1+x)^16/(3 - (1+x)^4)^5 + (1+x)^25/(3 - (1+x)^5)^6 + ...

%Y Cf. A301312.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 21 2018