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 1-3 of 3 results.

A220353 G.f.: Sum_{n>=0} (1 - (1-x)^n)^n.

Original entry on oeis.org

1, 1, 4, 23, 176, 1697, 19805, 271669, 4285195, 76430799, 1521161530, 33422603485, 803584699252, 20986514811397, 591616582807036, 17905570068475471, 579092313210791549, 19931241131544637637, 727395001560116046739, 28057672464546863483509, 1140566596105346550309751, 48735378037084078566334897, 2183719157723179429519093520, 102386962560815561519635957007
Offset: 0

Views

Author

Paul D. Hanna, Dec 11 2012

Keywords

Comments

Limit n->infinity A220353(n)/A187826(n) = 1. - Vaclav Kotesovec, Nov 08 2014

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 176*x^4 + 1697*x^5 + 19805*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x + (2*x - x^2)^2 + (3*x - 3*x^2 + x^3)^3 + (4*x - 6*x^2 + 4*x^3 - x^4)^4 + (5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^5 +...
(2) A(x) = (1-x) + (1-x)^2*(2*x - x^2) + (1-x)^3*(3*x - 3*x^2 + x^3)^2 + (1-x)^4*(4*x - 6*x^2 + 4*x^3 - x^4)^3 + (1-x)^5*(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^4 +...

Links

Crossrefs

Cf. A220352, A122400, A187827, A187826.

Programs

  • Mathematica
    terms = 24;
gf = 1 + Sum[(1 - (1 - x)^n)^n, {n, 1, terms}] + O[x]^terms;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 01 2018 *)
  • PARI
    {a(n)=local(q=1/(1-x+x*O(x^n)),A=1);A=sum(k=0,n,q^(-k^2)*(q^k-1)^k);polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(q=1/(1-x+x*O(x^n)),A=1);A=sum(k=1,n+1,q^(-k^2)*(q^k-1)^(k-1));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=1} (1-x)^n * (1 - (1-x)^n)^(n-1).
a(n) = c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.93418651575946259471737... . - Vaclav Kotesovec, May 06 2014
In closed form, c = 2^(log(2)/2-1) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015

Extensions

a(22)-a(23) corrected by Andrew Howroyd, Feb 22 2018

A187826 G.f. satisfies: A(x) = Sum_{n>=0} ((1 + x*A(x))^n - 1)^n / (1 + x*A(x))^(n^2).

Original entry on oeis.org

1, 1, 4, 26, 219, 2227, 26438, 359904, 5555201, 96383191, 1864908541, 39929905561, 938897407239, 24069888638463, 668309231078015, 19977542570492051, 639571311256259372, 21828488143257352752, 791044181963746918758, 30331001954496565907536
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2012

Keywords

Comments

Limit n->infinity A220353(n)/A187826(n) = 1. - Vaclav Kotesovec, Nov 08 2014

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 219*x^4 + 2227*x^5 + 26438*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x*A(x)/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)^2/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^3/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^4/(1+x*A(x))^16 +...
(2) A(x) = 1/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^2/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^3/(1+x*A(x))^16 +...

Links

Crossrefs

Cf. A220352, A187827, A220353.

Programs

  • PARI
    {a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = Sum_{n>=1} ((1+x*A(x))^n - 1)^(n-1) / (1+x*A(x))^(n^2).
a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 2.341658334181687683758... . - Vaclav Kotesovec, Nov 08 2014
In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1-log(2))/2)). - Vaclav Kotesovec, May 03 2015

A121886 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A122399(k).

Original entry on oeis.org

1, 1, 5, 40, 444, 6324, 110023, 2261576, 53632424, 1441341350, 43290170494, 1437020742408, 52243864528990, 2064488610832106, 88106523694973953, 4038627301344466648, 197888243609535940091, 10321811633042512528240
Offset: 0

Views

Author

Vladeta Jovovic, Aug 31 2006

Keywords

Comments

Number of square matrices with nonnegative integer entries and without zero rows such that sum of all entries is equal to n. - Vladeta Jovovic, Mar 04 2008

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 444*x^4 + 6324*x^5 +...
where
A(x) = 1 + (1/(1-x) - 1) + (1/(1-x)^2 - 1)^2 + (1/(1-x)^3 - 1)^3 + ...
Also,
A(x) = 1/2 + (1-x)/(1 + (1-x))^2 + (1-x)^2/(1 + (1-x)^2)^3 +  + (1-x)^3/(1 + (1-x)^3)^4 + (1-x)^4/(1 + (1-x)^4)^5 + ...

Crossrefs

Cf. A104209, A220352.

Programs

  • Mathematica
    Flatten[{1,Table[1/n!* Sum[Abs[StirlingS1[n,k]]*Sum[m^k*m!*StirlingS2[k, m], {m, 1, k}],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 07 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(1/(1-x+x*O(x^n))^m-1)^m),n)}

Formula

G.f.: Sum_{n>=0} ( 1/(1-x)^n - 1 )^n.
G.f.: Sum_{n>=0} (1-x)^n / (1 + (1-x)^n)^(n+1). - Paul D. Hanna, Sep 07 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.38377369607518184186200387319561108... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from Max Alekseyev, Feb 01 2007
Showing 1-3 of 3 results.

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