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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A257605 Expansion of g.f.: exp( Sum_{n>=1} A251661(n)*x^n/n ) where A251661(n) = Sum_{k=0..n} C(n, k) * (2^k + 3^k)^(n-k).

Original entry on oeis.org

1, 3, 12, 68, 606, 9438, 271154, 14272266, 1350900204, 226478780848, 67039275113982, 34862320055916606, 31905434621918041764, 51191148619374796495296, 144350180842362122992451022, 712487785268333349746955065478, 6171550949441004942637166827656834
Offset: 0

Views

Author

Paul D. Hanna, May 02 2015

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 68*x^3 + 606*x^4 + 9438*x^5 + 271154*x^6 + ...
The logarithm of g.f. A(x) begins:
log(A(x)) = 3*x + 15*x^2/2 + 123*x^3/3 + 1671*x^4/4 + 37863*x^5/5 + ... + A251661(n)*x^n/n + ...
which may be written as:
log(A(x)) = (2^1 + 1) * x +
(2^2 + 2*(2 + 3)^1 + 1) * x^2/2 +
(2^3 + 3*(2 + 3)^2 + 3*(2^2 + 3^2)^1 + 1) * x^3/3 +
(2^4 + 4*(2 + 3)^3 + 6*(2^2 + 3^2)^2 + 4*(2^3 + 3^3)^1 + 1) * x^4/4 +
(2^5 + 5*(2 + 3)^4 + 10*(2^2 + 3^2)^3 + 10*(2^3 + 3^3)^2 + 5*(2^4 + 3^4)^1 + 1) * x^5/5 + ...

Links

Crossrefs

Cf. A251661.

Programs

  • Mathematica
    A251661[n_]:= A251661[n]= Sum[Binomial[n,k]*(2^k +3^k)^(n-k), {k,0,n}];
With[{m = 50}, CoefficientList[Series[Exp[Sum[A251661[j]*x^j/j, {j, 2*m}]], {x, 0, m}], x]] (* G. C. Greubel, Mar 24 2022 *)
  • PARI
    {A251661(n) = sum(k=0, n, binomial(n, k) * (2^k + 3^k)^(n-k) )}
{a(n) = local(A = exp( sum(m=1,n,A251661(m)*x^m/m) +x*O(x^n)) );polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))
  • Sage
    m=40
@CachedFunction
def A251661(n): return sum( binomial(n, k)*(2^k + 3^k)^(n-k) for k in (0..n) )
def p(x): return exp( sum(A251661(j)*x^j/j for j in (1..2*m)) )
def A257605(n): return ( p(x) ).series(x, n+1).list()[n]
[A257605(n) for n in (0..m)] # G. C. Greubel, Mar 24 2022

Formula

G.f.: exp( Sum_{n>=1} A251661(n)*x^n/n ).

A251671 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^k.

Original entry on oeis.org

1, 6, 180, 43398, 88701816, 1573206748746, 248688444559874580, 356335498302585834118638, 4663871943514788530035646937456, 558720685051192771669885091319459750546, 612058892657175926094223171960469926874935754700
Offset: 0

Views

Author

Paul D. Hanna, Jan 21 2015

Keywords

Examples

			G.f.: A(x) = 1 + 6*x + 180*x^2 + 43398*x^3 + 88701816*x^4 + 1573206748746*x^5 +...
where A(x) = 1/(1-x) + (2+3)*x/(1-x)^2 + (2^2+3^2)*x^2/(1-x)^3 + (2^3+3^3)^3*x^3/(1-x)^4 +...
ILLUSTRATION OF INITIAL TERMS:
a(0) = 1*(2^0+3^0)^0 = 1;
a(1) = 1*(2^0+3^0)^0 + 1*(2^1+3^1)^1 = 6;
a(2) = 1*(2^0+3^0)^0 + 2*(2^1+3^1)^1 + 1*(2^2+3^2)^2 = 180;
a(3) = 1*(2^0+3^0)^0 + 3*(2^1+3^1)^1 + 3*(2^2+3^2)^2 + 1*(2^3+3^3)^3 = 43398;
a(4) = 1*(2^0+3^0)^0 + 4*(2^1+3^1)^1 + 6*(2^2+3^2)^2 + 4*(2^3+3^3)^3 + 1*(2^4+3^4)^4 = 88701816; ...

Crossrefs

Cf. A251661, A251184, A060722.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * (2^k + 3^k)^k,{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jan 25 2015 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k) * (2^k + 3^k)^k )}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=sum(m=0, n, (2^m + 3^m)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (2^n + 3^n)^n * x^n / (1-x)^(n+1).
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jan 25 2015
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