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

A364987 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^4.

Original entry on oeis.org

1, 1, 10, 183, 5140, 196005, 9468486, 554425963, 38171336680, 3022130473065, 270537702834250, 27021535857472431, 2979254055371578524, 359411244032212931533, 47093111659782104431438, 6660135357832421444841555, 1011181346455643980818939856
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A006153, A295238, A364983.
Cf. A002293, A349331.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k, k)/((3*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+1,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002293(k)/(n-k)!.
a(n) ~ 2*sqrt(1 + LambertW(27/256)) * n^(n-1) / (3^(3/2) * exp(n) * LambertW(27/256)^n). - Vaclav Kotesovec, Nov 11 2024

A364983 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^3.

Original entry on oeis.org

1, 1, 8, 111, 2332, 66125, 2368086, 102616759, 5222638856, 305436798009, 20186656927210, 1488021110087171, 121044207712073196, 10771321471267219525, 1040877104088653696606, 108549742436141933697135, 12151467262433697322437136, 1453367472748861203540942065
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364984, A364985, A364986.
Cf. A006153, A295238, A364987.
Cf. A001764, A307678.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (3*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A001764(k)/(n-k)!.
a(n) ~ sqrt(3) * sqrt(1 + LambertW(4/27)) * n^(n-1) / (2^(3/2) * exp(n) * LambertW(4/27)^n). - Vaclav Kotesovec, Nov 11 2024

A364982 E.g.f. satisfies A(x) = 1 + x*A(x)^2*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 6, 69, 1204, 28345, 842406, 30282385, 1278159240, 61979238513, 3395850105610, 207490382754721, 13989267347891628, 1031687145559176457, 82618837044274734126, 7139807492658000170865, 662286433378726179463696, 65635135687587192429274849
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A213644, A295238.
Cf. A161633, A364986, A364989.
Cf. A001764.

Programs

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

Formula

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

A377503 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^2.

Original entry on oeis.org

1, 2, 18, 270, 5936, 173330, 6335772, 278724362, 14350790064, 847007698338, 56397332340020, 4182866692785242, 342022887565717800, 30570009715185100082, 2965368922693150575084, 310276298423966343555690, 34834957115496822249510752, 4177193847524372747798263106
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377504, A377528.
Cf. A006013, A364983.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364983.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (k+1)*(n-k)! ).

A377504 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 36, 735, 21972, 871995, 43308378, 2588123811, 180990517032, 14507325973395, 1311719669172750, 132102208441613883, 14666354372331521676, 1779817542971018697003, 234399632982398657764578, 33297612755940733707395955, 5075234637265322738651060688, 826215756199826873368252279971
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377503, A377528.
Cf. A006632, A364987.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k+2, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364987.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+2,k)/( (k+1)*(n-k)! ).

A377526 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^5.

Original entry on oeis.org

1, 1, 12, 273, 9604, 460105, 27966126, 2062219117, 178897527768, 17853102321489, 2014988044093210, 253792946798597701, 35290880970687039732, 5370055269772474994713, 887591963820839894529654, 158357028389450319651183165, 30332317748593431632078480176, 6208425034878692992471996557217
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Comments

In general, for k > 1, if e.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^k, then a(n) ~ sqrt(k*(1 + LambertW((k-1)^(k-1)/k^k))) * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW((k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Nov 11 2024

Crossrefs

Cf. A006153, A295238, A364983, A364987.
Cf. A002294.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k, k)/((4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002294(k)/(n-k)!.
a(n) ~ sqrt(5*(1 + LambertW(256/3125))) * n^(n-1) / (8 * exp(n) * LambertW(256/3125)^n). - Vaclav Kotesovec, Nov 11 2024

A377528 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^4.

Original entry on oeis.org

1, 4, 60, 1548, 58456, 2930020, 183763704, 13866109012, 1224251041248, 123885272536452, 14140672597851880, 1797709847594145364, 251941291752251706576, 38593132701417704324356, 6415647343472197357272984, 1150373241484390263973203540, 221318733487356013660505462464
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377503, A377504.
Cf. A377526, A377527.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+3, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377526.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+3,k)/( (k+1)*(n-k)! ).

A381998 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^2.

Original entry on oeis.org

1, 1, 8, 90, 1472, 31920, 865152, 28197904, 1075122176, 46976064768, 2315080816640, 127068467480064, 7688296957870080, 508450036968779776, 36490818871396499456, 2824787199565881477120, 234622076533699738861568, 20813348299168251651883008, 1964063064959266899440959488
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A336950, A381997, A381999, A382000, A382001.
Cf. A000108, A295238, A379885.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A000108(k)/(n-k)!.
From Vaclav Kotesovec, Mar 22 2025: (Start)
E.g.f.: 2/(1 + sqrt(1 - 4*exp(2*x)*x)).
a(n) ~ sqrt(1 + LambertW(1/2)) * 2^(n + 1/2) * n^(n-1) / (exp(n) * LambertW(1/2)^n). (End)

A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4*x*exp(x))).

Original entry on oeis.org

1, -1, 2, -9, 68, -705, 9234, -146209, 2717000, -57986433, 1397949830, -37576332321, 1114326129564, -36141571087297, 1272713716466906, -48360394499269665, 1972269941821097744, -85929979225787811585, 3983422470176606823054, -195765982110500512129057
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A006531, A052895, A295238.

Programs

  • Maple
    a:=series(2/(1+sqrt(1+4*x*exp(x))),x=0,20): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 + 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(-1)^m (m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 19}]
  • PARI
    a(n) = n!*sum(k=0, n, (-1)^k*k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Oct 30 2024

Formula

E.g.f.: 1/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + ...))))), a continued fraction.
a(n) ~ sqrt(2*(1+LambertW(-1/4))) * n^(n-1) / (exp(n) * (LambertW(-1/4))^n). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} (-1)^k * k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Oct 30 2024

A295240 Expansion of e.g.f. 1/(1 - x*exp(x)/(1 - 2*x*exp(x)/(1 - 3*x*exp(x)/(1 - 4*x*exp(x)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 8, 129, 3748, 172385, 11541246, 1060864189, 128254619480, 19735654508577, 3766841223919930, 873355411013249021, 241783431463815426516, 78781867440446089479937, 29844928122224237593463270, 13007143530120743289176560125, 6462200434400107274026753685296
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A295238, A295241, A295242.

Programs

  • Mathematica
    nmax = 16; CoefficientList[Series[1/(1 + ContinuedFractionK[-k x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n + 1/2) / exp(2*n - 1/2). - Vaclav Kotesovec, Aug 09 2021
Showing 1-10 of 14 results. Next

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

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