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

%I #7 Jan 14 2019 09:19:55

%S 1,1,4,39,508,8651,180541,4462875,127461499,4129414609,149614338010,

%T 5994046983553,263101175224096,12555981779337615,647278321588763668,

%U 35845661666812566803,2122283542537445564169,133773419366606401021391,8943959013589398905563475,632203137717788438029869627,47105820660836320646061788567,3690081064592874994757245005533,303181494230752217882627389578352

%N G.f. Sum_{n>=0} ((1 + 2*x)^n - (1 + x)^n)^n.

%H Paul D. Hanna, <a href="/A323323/b323323.txt">Table of n, a(n) for n = 0..400</a>

%F a(n) ~ c * d^n * n^n, where d = 1.406505772593750511415058484128041501018... and c = 0.485030515627129100167196305487639128... - _Vaclav Kotesovec_, Jan 14 2019

%e G.f.: A(x) = 1 + x + 4*x^2 + 39*x^3 + 508*x^4 + 8651*x^5 + 180541*x^6 + 4462875*x^7 + 127461499*x^8 + 4129414609*x^9 + 149614338010*x^10 + ...

%e such that

%e A(x) = 1 + ((1+2*x)-(1+x)) + ((1+2*x)^2-(1+x)^2)^2 + ((1+2*x)^3-(1+x)^3)^3 + ((1+2*x)^4-(1+x)^4)^4 + ((1+2*x)^5-(1+x)^5)^5 + ...

%o (PARI) {a(n) = my(A=1,X=x+x*O(x^n)); A = sum(m=0,n, ((1 + 2*X)^m - (1 + X)^m)^m); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 11 2019