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.

A365183 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 5, 34, 268, 2299, 20838, 196326, 1903524, 18868861, 190356231, 1948055058, 20173907384, 211020478270, 2226243632838, 23660868061422, 253099278807684, 2722819049879436, 29439894433161189, 319749417998303470, 3486914150183526920
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Links

Crossrefs

Cf. A002294, A365178, A365180, A365181, A365182.
Cf. A006605, A255673, A365189.
Cf. A364989.

Programs

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

Formula

a(n) = (1/(4*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(4*n+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)!.

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)!.

A365177 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 10, 201, 6220, 261465, 13925286, 898994383, 68240292856, 5956670911041, 587896878021130, 64738492669538391, 7869297152389747284, 1046629627952327990545, 151192146681811716344878, 23573456446401808474471455, 3945806733850334447131941616
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A364987, A364989, A365175, A365176.

Programs

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

Formula

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

A365175 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)).

Original entry on oeis.org

1, 1, 10, 189, 5476, 215145, 10701006, 644909503, 45687408712, 3721382812305, 342689189598010, 35206864089944151, 3992473080042706524, 495361299387667990537, 66752437447119717428422, 9708649781691227748131535, 1515863453268825963300368656
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A364987, A364989, A365176, A365177.
Cf. A161633, A213644, A364984.

Programs

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

Formula

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

A365176 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 10, 195, 5836, 236925, 12177966, 758458603, 55528414264, 4674208189977, 444823048027450, 47227542351423951, 5534636939373353604, 709653811287800826421, 98825110036657191358822, 14853654178825132742729715, 2396666529204491489278153456
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A364987, A364989, A365175, A365177.

Programs

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

Formula

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

A377630 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^4 ).

Original entry on oeis.org

1, 4, 52, 1212, 41512, 1889700, 107684664, 7384011796, 592485333472, 54488274328836, 5652345176418280, 653054114586249684, 83175314479016845584, 11578838832843098353732, 1749242011108507789948312, 285034599164755404426493140, 49833544890911336997795542464
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Links

Crossrefs

Cf. A161633, A377553, A377554.
Cf. A364989, A377633.

Programs

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

Formula

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A364989.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(4*n+4,k)/(n-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)! ).

A377633 E.g.f. satisfies A(x) = (1 + x * A(x)^2 * exp(x*A(x)^2))^2.

Original entry on oeis.org

1, 2, 22, 474, 15512, 685450, 38251212, 2582466950, 204744089104, 18650699228754, 1919767092675860, 220378941471652942, 27916112634179586456, 3868191824049865945178, 582034397153512353488284, 94509039130326185624148150, 16472375319790734221438146592, 3067435511995844132675459958178
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A364989, A377630.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364989.
a(n) = (n!/(2*n+1)) * Sum_{k=0..n} k^(n-k) * binomial(4*n+2,k)/(n-k)!.
Showing 1-9 of 9 results.

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

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