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

A220352 G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n^2).

Original entry on oeis.org

1, 1, 3, 16, 118, 1116, 12869, 175096, 2745726, 48756438, 967026762, 21188546616, 508286084222, 13249410224210, 372908807794347, 11270832179901016, 364083312029454453, 12518063823862065816, 456432182550333723335, 17591590487681007523476
Offset: 0

Views

Author

Paul D. Hanna, Dec 11 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1116*x^5 + 12869*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x/(1+x) + (2*x + x^2)^2/(1+x)^4 + (3*x + 3*x^2 + x^3)^3/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^4/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^5/(1+x)^25 +...
(2) A(x) = 1/(1+x) + (2*x + x^2)/(1+x)^4 + (3*x + 3*x^2 + x^3)^2/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^3/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^4/(1+x)^25 +...

Links

Crossrefs

Cf. A220353, A121886, A187826.

Programs

  • PARI
    {a(n)=local(q=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=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(q=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=0,20,print1(a(n),", "))

Formula

G.f.: Sum_{n>=1} ((1+x)^n - 1)^(n-1) / (1+x)^(n^2).
a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 1.44832302735058524286860126583754380692... . - 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

A187827 G.f. satisfies: A(x) = Sum_{n>=0} (1 - (1 - x*A(x))^n)^n.

Original entry on oeis.org

1, 1, 5, 36, 325, 3468, 42519, 590268, 9201740, 160150252, 3095440553, 66068011710, 1547572760559, 39529002357409, 1094096683131616, 32622859912512090, 1042350065213470532, 35521574976088978133, 1285782300453328211074, 49256935742079848796102
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 36*x^3 + 325*x^4 + 3468*x^5 + 42519*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x*A(x) + (1 - (1-x*A(x))^2)^2 + (1 - (1-x*A(x))^3)^3 + (1 - (1-x*A(x))^4)^4 + (1 - (1-x*A(x))^5)^5 +...
(2) A(x) = (1-x*A(x)) + (1-x*A(x))^2*(1 - (1-x*A(x))^2) + (1-x*A(x))^3*(1 - (1-x*A(x))^3)^2 + (1-x*A(x))^4*(1 - (1-x*A(x))^4)^3 + (1-x)^5*(1 - (1-x*A(x))^5)^4 +...

Links

Crossrefs

Cf. A220353, A187826.

Programs

  • PARI
    {a(n)=local(q, A=1); for(i=1,n,q=1/(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=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(q, A=1); for(i=1,n,q=1/(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=0,20,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = Sum_{n>=1} (1-x*A(x))^n * (1 - (1-x*A(x))^n)^(n-1).
a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 3.7860088... . - Vaclav Kotesovec, Nov 08 2014
Showing 1-3 of 3 results.

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