A337851 a(n) = (2^n + 2)^n.
1, 4, 36, 1000, 104976, 45435424, 82653950016, 627485170000000, 19631688197463081216, 2504194578379511247798784, 1292628144912333835229805413376, 2687153475176994340820312500000000000, 22431765115399782718874449007331506546282496
Offset: 0
Keywords
Examples
O.g.f.: A(x) = 1 + 4*x + 36*x^2 + 1000*x^3 + 104976*x^4 + 45435424*x^5 + 82653950016*x^6 + 627485170000000*x^7 + 19631688197463081216*x^8 + ... where A(x) = 1/(1 - 2*x) + 2*x/(1 - 2^2*x)^2 + 2^4*x^2/(1 - 2^3*x)^3 + 2^9*x^3/(1 - 2^4*x)^4 + 2^16*x^4/(1 - 2^5*x)^5 + 2^25*x^5/(1 - 2^6*x)^6 + ...
Programs
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PARI
{a(n,q,m,b) = (m*q^n + b)^n} for(n=0,15, print1(a(n,q=2,m=1,b=2),", ")) -
PARI
/* E.g.f. formula: */ {a(n,q,m,b) = polcoeff( sum(k=0,n, m^k * q^(k^2) * x^k / (1 - b*q^k*x +x*O(x^n))^(k+1)), n)} for(n=0,15, print1(a(n,q=2,m=1,b=2),", ")) -
PARI
/* E.g.f. formula: */ {a(n,q,m,b) = n! * polcoeff( sum(k=0,n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)} for(n=0,15, print1(a(n,q=2,m=1,b=2),", "))
Formula
O.g.f.: Sum_{n>=0} 2^(n^2) * x^n/(1 - 2^(n+1)*x)^(n+1) = Sum_{n>=0} (2^n + 2)^n * x^n.
E.g.f.: Sum_{n>=0} 2^(n^2) * exp(2^(n+1)*x) * x^n / n! = Sum_{n>=0} (2^n + 2)^n * x^n / n!.
a(n) = 2^n * A165327(n) for n >= 0.
Comments