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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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A251184 a(n) = Sum_{k=0..n} binomial(n,k) * (2^k + 3)^k.

Original entry on oeis.org

1, 6, 60, 1494, 135960, 53187306, 90775495620, 662696938144254, 20254044105203565360, 2548567387213968842106066, 1305307745923414524904985640540, 2701621925224675918174411993192242534, 22497571753504120612543889146545487252194120, 751859379104473999264213367292907652977053019173306
Offset: 0

Views

Author

Paul D. Hanna, Jan 20 2015

Keywords

Examples

			G.f.: A(x) = 1 + 6*x + 60*x^2 + 1494*x^3 + 135960*x^4 + 53187306*x^5 +...
where we have the identity:
(1) A(x) = 1/(1-x) + (2+3)*x/(1-x)^2 + (2^2+3)^2*x^2/(1-x)^3 + (2^3+3)^3*x^3/(1-x)^4 + (2^4+3)^4*x^4/(1-x)^5 + (2^5+3)^5*x^5/(1-x)^6 +...
(2) A(x) = 1/(1-4*x) + 2*x/(1-x - 3*2*x)^2 + 2^4*x^2/(1-x - 3*2^2*x)^3 + 2^9*x^3/(1-x - 3*2^3*x)^4 + 2^16*x^4/(1-x - 3*2^4*x)^5 + 2^25*x^5/(1-x - 3*2^5*x)^6 +...
Illustration of initial terms.
a(0) = 1;
a(1) = 1 + (2+3) = 6;
a(2) = 1 + 2*(2+3) + (2^2+3)^2 = 60;
a(3) = 1 + 3*(2+3) + 3*(2^2+3)^2 + (2^3+3)^3 = 1494;
a(4) = 1 + 4*(2+3) + 6*(2^2+3)^2 + 4*(2^3+3)^3 + (2^4+3)^4 = 135960;
a(5) = 1 + 5*(2+3) + 10*(2^2+3)^2 + 10*(2^3+3)^3 + 5*(2^4+3)^4 + (2^5+3)^5 = 53187306; ...

Crossrefs

Cf. A251182, A251183.

Programs

  • PARI
    {a(n)=sum(k=0, n, binomial(n, k) * (2^k + 3)^k )}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=sum(m=0, n, (2^m + 3)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=sum(m=0, n, 2^(m^2) * x^m / (1-x - 3*2^m*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 * x^n / (1-x)^(n+1).
G.f.: Sum_{n>=0} 2^(n^2) * x^n / (1-x - 3*2^n*x)^(n+1).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jan 25 2015

A251183 a(n) = Sum_{k=0..n} binomial(n,k) * (-1)^(n-k) * (2^k + 1)^k.

Original entry on oeis.org

1, 2, 20, 662, 80744, 38724842, 75185316860, 593940189201662, 19026394370724334544, 2460515237682283759296722, 1280059938187721964276628432100, 2672755638642586172344574379667166822, 22366135140308331991861876017552822964355384, 749476887752252949738787898479359344347670538150842
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2015

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 20*x^2 + 662*x^3 + 80744*x^4 + 38724842*x^5 +...
where we have the identity:
(1) A(x) = 1/(1-x) + (2+1)*x/(1+x)^2 + (2^2+1)^2*x^2/(1+x)^3 + (2^3+1)^3*x^3/(1+x)^4 + (2^4+1)^4*x^4/(1+x)^5 + (2^5+1)^5*x^5/(1+x)^6 +...
(2) A(x) = 1 + 2*x/(1+x - 2*x)^2 + 2^4*x^2/(1+x - 2^2*x)^3 + 2^9*x^3/(1+x - 2^3*x)^4 + 2^16*x^4/(1+x - 2^4*x)^5 + 2^25*x^5/(1+x - 2^5*x)^6 +...
Illustration of initial terms:
a(0) = 1;
a(1) = -1 + (2+1) = 2;
a(2) = 1 - 2*(2+1) + (2^2+1)^2 = 20;
a(3) = -1 + 3*(2+1) - 3*(2^2+1)^2 + (2^3+1)^3 = 662;
a(4) = 1 - 4*(2+1) + 6*(2^2+1)^2 - 4*(2^3+1)^3 + (2^4+1)^4 = 80744;
a(5) = -1 + 5*(2+1) - 10*(2^2+1)^2 + 10*(2^3+1)^3 - 5*(2^4+1)^4 + (2^5+1)^5 = 38724842; ...

Crossrefs

Cf. A251182, A251184.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * (-1)^(n-k) * (2^k + 1)^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 25 2015 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k) * (-1)^(n-k) * (2^k + 1)^k )}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=sum(m=0, n, (2^m + 1)^m * x^m / (1+x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=sum(m=0, n, 2^(m^2) * x^m / (1+x - 2^m*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 + 1)^n * x^n / (1+x)^(n+1).
G.f.: Sum_{n>=0} 2^(n^2) * x^n / (1+x - 2^n*x)^(n+1).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jan 25 2015
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