cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A244755 a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k).

Original entry on oeis.org

1, 3, 13, 87, 985, 19563, 697573, 44195007, 4985202865, 987432857043, 344306650353853, 209169206074748967, 222262777197258910345, 409907753371580011362363, 1317924525238880964004945813, 7341603216747343890845790989967, 71176841502529490992224798115792225
Offset: 0

Views

Author

Paul D. Hanna, Jul 05 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + 3*x + 13*x^2/2! + 87*x^3/3! + 985*x^4/4! + 19563*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+3^0)^1 + (1+3^1)^0 = 3;
a(2) = (1+3^0)^2 + 2*(1+3^1)^1 + (1+3^2)^0 = 13;
a(3) = (1+3^0)^3 + 3*(1+3^1)^2 + 3*(1+3^2)^1 + (1+3^3)^0 = 87;
a(4) = (1+3^0)^4 + 4*(1+3^1)^3 + 6*(1+3^2)^2 + 4*(1+3^3)^1 + (1+3^4)^0 = 985; ...

Links

Crossrefs

Cf. A244754, A244756, A244760, A135079, A243918.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * (1 + 3^k)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 25 2015 *)
  • PARI
    {a(n) = sum(k=0,n,binomial(n,k) * (1 + 3^k)^(n-k) )}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* E.g.f. Sum_{n>=0} exp((1+3^n)*x)*x^n/n!" */
{a(n)=n!*polcoeff(sum(k=0, n, exp((1+3^k)*x +x*O(x^n))*x^k/k!), n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* O.g.f. Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(1+3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} exp((1+3^n)*x) * x^n/n!.
O.g.f.: Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1).
a(n) ~ c * 3^(n^2/4) * 2^(n+1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd. - Vaclav Kotesovec, Jan 25 2015

A244756 a(n) = Sum_{k=0..n} C(n,k) * (2 + 3^k)^(n-k).

Original entry on oeis.org

1, 4, 20, 136, 1424, 25504, 831680, 49526656, 5359464704, 1033951896064, 354410768092160, 213011725510260736, 224795751647646224384, 412813583857427719266304, 1323683536183041967893954560, 7361415226356149639592083685376, 71294465534894253722438522191806464
Offset: 0

Views

Author

Paul D. Hanna, Jul 05 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 20*x^2/2! + 136*x^3/3! + 1424*x^4/4! + 25504*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (2+3^0)^1 + (2+3^1)^0 = 4;
a(2) = (2+3^0)^2 + 2*(2+3^1)^1 + (2+3^2)^0 = 20;
a(3) = (2+3^0)^3 + 3*(2+3^1)^2 + 3*(2+3^2)^1 + (2+3^3)^0 = 136;
a(4) = (2+3^0)^4 + 4*(2+3^1)^3 + 6*(2+3^2)^2 + 4*(2+3^3)^1 + (2+3^4)^0 = 1424; ...

Links

Crossrefs

Cf. A244755, A244760, A244754, A243918.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * (2 + 3^k)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 25 2015 *)
  • PARI
    {a(n) = sum(k=0,n,binomial(n,k) * (2 + 3^k)^(n-k) )}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* E.g.f. Sum_{n>=0} exp((2+3^n)*x)*x^n/n!" */
{a(n)=n!*polcoeff(sum(k=0, n, exp((2+3^k)*x +x*O(x^n))*x^k/k!), n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* O.g.f. Sum_{n>=0} x^n/(1 - (2+3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2+3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} exp((2+3^n)*x) * x^n/n!.
O.g.f.: Sum_{n>=0} x^n/(1 - (2+3^n)*x)^(n+1).
a(n) ~ c * 3^(n^2/4) * 2^(n+1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd. - Vaclav Kotesovec, Jan 25 2015

A244760 a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k) * 2^k.

Original entry on oeis.org

1, 4, 24, 232, 3840, 111904, 5785344, 529662592, 85449338880, 24204383609344, 11986829259362304, 10361640102119729152, 15589910824599107174400, 40815393380277274447519744, 185575767151388880816233447424, 1465910356757779350231777997914112
Offset: 0

Views

Author

Paul D. Hanna, Jul 05 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3840*x^4/4! + 111904*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+3^0)^1 + (1+3^1)^0*2 = 4;
a(2) = (1+3^0)^2 + 2*(1+3^1)^1*2 + (1+3^2)^0*2^2 = 24;
a(3) = (1+3^0)^3 + 3*(1+3^1)^2*2 + 3*(1+3^2)^1*2^2 + (1+3^3)^0*2^3 = 232;
a(4) = (1+3^0)^4 + 4*(1+3^1)^3*2 + 6*(1+3^2)^2*2^2 + 4*(1+3^3)^1*2^3 + (1+3^4)^0*2^4 = 3480; ...

Links

Crossrefs

Cf. A244755, A244756, A244754, A243918.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * (1 + 3^k)^(n-k) * 2^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 25 2015 *)
  • PARI
    {a(n) = sum(k=0,n,binomial(n,k) * (1 + 3^k)^(n-k)*2^k )}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* E.g.f. Sum_{n>=0} exp((1+3^n)*x)*(2*x)^n/n! */
{a(n)=n!*polcoeff(sum(k=0, n, exp((1+3^k)*x +x*O(x^n))*(2*x)^k/k!), n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* O.g.f. Sum_{n>=0} (2*x)^n/(1 - (1+3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, (2*x)^k/(1-(1+3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} exp((1+3^n)*x) * (2*x)^n/n!.
O.g.f.: Sum_{n>=0} (2*x)^n/(1 - (1+3^n)*x)^(n+1).
a(n) ~ c * 3^(n^2/4) * 2^((3*n+1)/2) / sqrt(Pi*n), where c = sum_{k=-inf..+inf} 1/(3^(k^2) * 2^k) = 1.88621563508001862566062... if n is even, and c = sum_{k=-inf..+inf} 1/(3^((k+1/2)^2) * 2^(k+1/2)) = 1.88659407336643412717014... if n is odd. - Vaclav Kotesovec, Jan 25 2015

A251661 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^(n-k).

Original entry on oeis.org

1, 3, 15, 123, 1671, 37863, 1447515, 93965763, 10456301871, 2001375249423, 663553617119475, 381265346343864843, 381607689867265672551, 664239239404717367975223, 2018751151993358704057734795, 10680818706115450217386068210963, 98710608829560784063971722066895711
Offset: 0

Views

Author

Paul D. Hanna, Jan 21 2015

Keywords

Comments

a(n) == 0 (mod 3) for n>0. - Paul D. Hanna, May 02 2015

Examples

			E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 123*x^3/3! + 1671*x^4/4! + 37863*x^5/5! +...
where
A(x) = exp(2*x) + exp(5*x)*x + exp(13*x)*x^2/2! + exp(35*x)*x^3/3! + exp(97*x)*x^4/4! + exp(275*x)*x^5/5! + exp(793*x)*x^6/6! +...+ exp((2^n+3^n)*x)*x^n/n! +...
ILLUSTRATION OF INITIAL TERMS:
a(0) = 1*(2^0+3^0)^0 = 1;
a(1) = 1*(2^0+3^0)^1 + 1*(2^1+3^1)^0 = 3;
a(2) = 1*(2^0+3^0)^2 + 2*(2^1+3^1)^1 + 1*(2^2+3^2)^0 = 15;
a(3) = 1*(2^0+3^0)^3 + 3*(2^1+3^1)^2 + 3*(2^2+3^2)^1 + 1*(2^3+3^3)^0 = 123;
a(4) = 1*(2^0+3^0)^4 + 4*(2^1+3^1)^3 + 6*(2^2+3^2)^2 + 4*(2^3+3^3)^1 + 1*(2^4+3^4)^0 = 1671; ...

Links

Crossrefs

Cf. A244754, A251671.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * (2^k + 3^k)^(n-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)^(n-k) )}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} exp((2^n + 3^n)*x)*x^n/n!: */
{a(n)=n!*polcoeff(sum(k=0, n, exp((2^k + 3^k)*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* O.g.f. Sum_{n>=0} x^n/(1 - (2^n + 3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k + 3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f.: Sum_{n>=0} exp((2^n + 3^n)*x) * x^n/n!.
O.g.f.: Sum_{n>=0} x^n/(1 - (2^n + 3^n)*x)^(n+1).
a(n) ~ c * 3^(n^2/4) * 2^(n+1/2) / sqrt(Pi*n), where c = JacobiTheta3(0, 1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0, 1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd. - Vaclav Kotesovec, Jan 25 2015
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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