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 1-10 of 14 results. Next

A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

Original entry on oeis.org

1, 1, 3, 19, 181, 2321, 37501, 731935, 16758393, 440525377, 13077834841, 432796650551, 15799794395749, 630773263606513, 27339525297079269, 1278550150117141231, 64171287394646697841, 3440711053857464325377
Offset: 0

Views

Author

Paul D. Hanna, Jun 17 2009

Keywords

Examples

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...

Links

Crossrefs

Cf. A161633 (e.g.f. = log(A(x))/x).
Cf. A212722, A212917, A245265, A125500.

Programs

  • Mathematica
    Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(n-1,n-k)))}
  • PARI
    {a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)));n!*polcoeff(A^m,n)}

Formula

a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).
E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - Paul D. Hanna, Aug 02 2013
a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

A212917 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 31, 469, 9681, 254701, 8131999, 305626329, 13218345793, 646712664121, 35315446759671, 2129341219106773, 140506900034640049, 10071589943109973461, 779311468200041101711, 64742128053980794659121, 5747587082198264156035329, 543023929087191507383612785
Offset: 0

Views

Author

Paul D. Hanna, May 30 2012

Keywords

Comments

From Vaclav Kotesovec, Jul 15 2014: (Start)
Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).
(End)

Examples

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^3 + x^2*A(x)^6 + x^3*A(x)^9 + x^4*A(x)^12 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^3) = x + 2*x^2/2! + 24*x^3/3! + 348*x^4/4! + 7140*x^5/5! + 186750*x^6/6! + ... + n*A366233(n-1)*x^n/n! + ...
A(x)^3 = 1 + 3*x + 15*x^2/2! + 153*x^3/3! + 2421*x^4/4! + 51363*x^5/5! + 1375029*x^6/6! + ...
A(x)^6 = 1 + 6*x + 48*x^2/2! + 576*x^3/3! + 9864*x^4/4! + 221256*x^5/5! + 6156756*x^6/6! + ...

Links

Crossrefs

Cf. A161630, A212722, A245265.
Cf. A366233 (log).

Programs

  • Mathematica
    Table[Sum[n! * (1 + 3*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)
  • PARI
    {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+3*(n-k))^(k-1)*binomial(n-1, n-k)))}
  • PARI
    {a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^3+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0, 21, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} n! * (1 + 3*(n-k))^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(m + 3*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ 3^(n-2/3) * n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2/3) * exp(n) * c^(2*n+1/6)), where c = LambertW(sqrt(3)/2) = 0.5166154518588324282494... . - Vaclav Kotesovec, Jul 15 2014

A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(2*x*A(x)).

Original entry on oeis.org

1, 1, 6, 54, 728, 13000, 290352, 7800016, 245115264, 8826560640, 358463525120, 16212238054144, 808215885708288, 44035925223746560, 2603618739995621376, 166031767704180111360, 11359670347331723952128, 830065763154102656204800, 64518486557995327748898816
Offset: 0

Views

Author

Paul D. Hanna, Oct 05 2023

Keywords

Comments

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum_{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum_{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.

Examples

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 54*x^3/3! + 728*x^4/4! + 13000*x^5/5! + 290352*x^6/6! + 7800016*x^7/7! + 245115264*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(2*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+1*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(-0*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-1*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-2*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-3*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-4*x*A(x))/6! + ...
and
A(x) = 1 + 3*1*3^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 3*2*4^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 3*3*5^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 3*4*6^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 3*5*7^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ... + A212722(n)*x^n/n! + ...

Crossrefs

Cf. A161633, A366233, A366234, A366235.
Cf. A365772 (dual), A212722 (exp(x*A(x))).

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(2*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)
  • PARI
    /* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 2^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(2*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+1, n-k)/(n+1) * 2^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 2^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(2*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(2*x)) ).
(3) A( x/(1 + x*exp(2*x)) ) = 1 + x*exp(2*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-2)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).
a(n) ~ n^(n-1) * (1 + 2*LambertW(1/sqrt(2)))^(n + 3/2) / (sqrt(1 + LambertW(1/sqrt(2))) * 2^(n+2) * exp(n) * LambertW(1/sqrt(2))^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A361093 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^2) - 1 ).

Original entry on oeis.org

1, 1, 7, 97, 2049, 58541, 2114143, 92419965, 4746108769, 280105517881, 18683156508471, 1389960074426969, 114119472522112225, 10249863809271551973, 999746622121255094479, 105236583967331849218741, 11891012005206169120252737, 1435560112909007680593616625
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A052873, A361094, A361095, A361096, A361097.
Cf. A212722, A361065.

Programs

  • Mathematica
    Table[n! * Sum[(2*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(n-1,n-k)/k!.
a(n) ~ n^(n-1) / (2 * 3^(1/4) * (2 - sqrt(3))^n * exp((2 - sqrt(3))*n - (sqrt(3) - 1)/2)). - Vaclav Kotesovec, Mar 02 2023

A245265 E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).

Original entry on oeis.org

1, 1, 3, 37, 649, 15461, 471571, 17456041, 760880625, 38178439849, 2167446089251, 137359883836781, 9612722107574521, 736277501363180557, 61265207586681046131, 5503291392884323494961, 530778414439201798454881, 54706967800114521799571921, 6000952913613549583603208515
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 15 2014

Keywords

Comments

Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ...

Links

Crossrefs

Cf. A161630 (p=1), A212722 (p=2), A212917 (p=3).
Cf. A030178.
Cf. A366234 (log).

Programs

  • Mathematica
    Table[Sum[n! * (1 + 4*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}]
  • PARI
    for(n=0,30, print1(sum(k=0,n, n!*(1 + 4*(n-k))^(k-1)/k!*binomial(n-1,n-k)), ", ")) \\ G. C. Greubel, Nov 17 2017

Formula

a(n) = Sum_{k=0..n} n! * (1 + 4*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ n^(n-1) * (1+2*LambertW(1))^(n+1/2) / (exp(n) * (LambertW(1))^(2*n) * (4*sqrt(1+LambertW(1)))). - Vaclav Kotesovec, Jul 15 2014

A361090 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)) ).

Original entry on oeis.org

1, 1, 3, 7, -11, -239, -179, 24991, 192025, -3955391, -89483399, 552615031, 46231717621, 254468241457, -26683006147979, -571848064714289, 14926049610344881, 825004339886219521, -2973711136010539535, -1134313888244827421465, -17734152216328857754739
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A161630, A212722, A212917, A361091, A361092.
Cf. A361067, A361095.

Programs

  • PARI
    a(n) = if(n==0, 1, n!*sum(k=1, n, (-n+k+1)^(k-1)*binomial(n-1, n-k)/k!));

Formula

a(n) = n! * Sum_{k=1..n} (-n+k+1)^(k-1) * binomial(n-1,n-k)/k! for n>0.

A361091 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 1, -71, -19, 10051, 12349, -3185391, -9346247, 1797304771, 9717361721, -1582301193527, -13722004186331, 2000705907453891, 25552516703201461, -3432004488804778079, -60960914621687232271, 7660860906885122096515
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A161630, A212722, A212917, A361090, A361092.
Cf. A361068, A361096.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-2*n+2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

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

A361092 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^3) ).

Original entry on oeis.org

1, 1, 3, -5, -107, 1041, 20701, -440033, -8464455, 343190593, 5639857561, -423764450889, -4968055259771, 754544622295153, 3846355902999429, -1818148417882379729, 6637679490204153841, 5658469355898945338625, -84578525845602646639823
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A161630, A212722, A212917, A361090, A361091.
Cf. A361069, A361097.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-3*n+3*k+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (-3*n+3*k+1)^(k-1) * binomial(n-1,n-k)/k!.

A382031 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x)^2)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 43, 1177, 46681, 2419291, 154587427, 11735209585, 1031418915121, 102979800567091, 11510663862332251, 1423811747933017609, 193073662118499898633, 28479005472094048953355, 4539456019668776334683731, 777538096585429376795405281, 142419954152382631361835929185
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A212722, A382029, A382030.
Cf. A380513, A382016.
Cf. A002293, A382044.

Programs

  • PARI
    a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+3*k, k)/((n+3*k)*(n-k-1)!)));

Formula

Let F(x) be the e.g.f. of A382044. F(x) = log(A(x))/x = B(x*A(x)^2).
a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+3*k,k)/((n+3*k) * (n-k-1)!) for n > 0.

A361143 E.g.f. satisfies A(x) = exp( x*A(x)^4/(1 - x*A(x)^2) ).

Original entry on oeis.org

1, 1, 11, 241, 8105, 370061, 21403675, 1500521485, 123685912817, 11724012791929, 1256517775425131, 150254377493878505, 19833528195709809817, 2864566162751107839493, 449364739762263286489403, 76084967168410028438252101, 13829896583435315152843525985
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2023

Keywords

Links

Crossrefs

Cf. A000262, A361142.
Cf. A212722, A361065, A361093.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*n+2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
Showing 1-10 of 14 results. Next

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