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 1-9 of 9 results.

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

A364984 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)).

Original entry on oeis.org

1, 1, 8, 117, 2596, 77705, 2936406, 134228059, 7204913528, 444331053873, 30963240318250, 2406301353714731, 206354828717754036, 19357367027097743449, 1971809610601104110942, 216754216326949771274715, 25575749384428387961718256, 3224227609551980271408565985
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364983, A364985, A364986.
Cf. A108447.

Programs

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

Formula

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

A364986 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 8, 129, 3196, 107465, 4575966, 236120059, 14322901832, 998966928897, 78770826493210, 6929685905371691, 672900446143476156, 71491442785783506577, 8249400210035835040022, 1027394346436911560475915, 137360293432089585554830096
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364983, A364984, A364985.
Cf. A161633, A364982, A364989.
Cf. A002293.

Programs

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

Formula

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

A377546 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^2 ).

Original entry on oeis.org

1, 2, 18, 294, 7136, 231410, 9421932, 462459242, 26593896912, 1754278123266, 130611457831700, 10835721949072922, 991315043401627320, 99154012317212577218, 10765112531819005907484, 1260860266373297376720810, 158473050112495481401395872, 21275613503385328981848681986
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Links

Crossrefs

Cf. A213644, A377548.
Cf. A364985.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*exp(x))^2)/x))
  • PARI
    a(n) = 2*n!*sum(k=0, n, k^(n-k)*binomial(2*n+k+2, k)/((2*n+k+2)*(n-k)!));

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364985.
a(n) = 2 * n! * Sum_{k=0..n} k^(n-k) * binomial(2*n+k+2,k)/( (2*n+k+2)*(n-k)! ).

A379867 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^2) - x*A(x)^2).

Original entry on oeis.org

1, 2, 23, 529, 18589, 884281, 53195407, 3874595089, 331580316473, 32614443047521, 3625839880813171, 449629404853604185, 61535275741655857621, 9213155228282408405185, 1498018121369750569371959, 262869047482982449625840161, 49515850496472530668242845041
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Cf. A377890, A379870.
Cf. A377892, A379884.
Cf. A377891, A377893, A379886.
Cf. A364985, A379864.

Programs

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

Formula

E.g.f.: sqrt( (1/x) * Series_Reversion( x * (exp(-x) - x)^2 ) ).
a(n) = n! * Sum_{k=0..n} (3*n-k+1)^(k-1) * binomial(3*n-k+1,n-k)/k!.

A377631 E.g.f. satisfies A(x) = 1/(1 - x * A(x)^4 * exp(x*A(x)^4)).

Original entry on oeis.org

1, 1, 12, 297, 11380, 593785, 39304206, 3155996557, 298106913336, 32391139027185, 3980284376962330, 545806093612966021, 82628400115183659012, 13688201250584241332809, 2463065653446247669021398, 478399017659163635014545405, 99757368661138669886988396016
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A377629, A377632.
Cf. A213644, A364985, A365177.
Cf. A364989.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*n+k+1,k)/( (4*n+k+1)*(n-k)! ).

A381479 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cos(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 6, 69, 1200, 28085, 828240, 29502473, 1232606592, 59114482569, 3201204188160, 193215861134989, 12862437022076928, 936256855741871677, 73978404781917941760, 6306254322850544942865, 576881179288397985054720, 56369243043268551691136657, 5859726074013471622734938112
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A185951, A364985, A381388.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+k+1, k)/(2*n+k+1)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n+k+1,k)/(2*n+k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.
E.g.f.: ( (1/x) * Series_Reversion( x*(1 - x*cos(x))^2 ) )^(1/2).

A377549 E.g.f. satisfies A(x) = 1 + x*A(x)^5*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 12, 285, 10444, 520465, 32882406, 2519264797, 227003238792, 23526134771553, 2757165645132010, 360564513170510341, 52053350012338720332, 8222888925567102799441, 1410913077291231960911934, 261306906300110395598900685, 51955790654759866661097707536
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Crossrefs

Cf. A364980, A364982, A364985, A365176.
Cf. A377547.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*n+3*k+1,k)/( (2*n+3*k+1)*(n-k)! ).

A381477 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cosh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 6, 75, 1440, 37445, 1231440, 49037527, 2294425728, 123393443049, 7500623201280, 508577491719011, 38057966976387072, 3115680296111519149, 277005128553759191040, 26579020362900758232495, 2737628961211699538657280, 301278578823933606439917137, 35281158151116225085977526272
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A185951, A364985, A381386.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+k+1, k)/(2*n+k+1)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n+k+1,k)/(2*n+k+1) * A185951(n,k).
E.g.f.: ( (1/x) * Series_Reversion( x*(1 - x*cosh(x))^2 ) )^(1/2).
Showing 1-9 of 9 results.

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

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