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 1-7 of 7 results.

A380663 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)) ).

Original entry on oeis.org

1, 2, 15, 208, 4285, 117936, 4075099, 169736960, 8282604537, 463604723200, 29287449579751, 2061571190059008, 160023548976361525, 13580237335641417728, 1250935473495646861875, 124307671411309327876096, 13255531892787507819759601, 1509841440567809574906101760
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377831, A380664.
Cf. A052873, A380665.

Programs

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

Formula

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

A380666 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x/(1 - x)^2) ).

Original entry on oeis.org

1, 3, 33, 670, 20193, 812736, 41056921, 2499780144, 178288822305, 14584953692800, 1346528845766481, 138513476506770432, 15711724851356153857, 1948422564510092267520, 262263690685637016402825, 38082186820362623941236736, 5933845220766237850177220289, 987599486681637240983472930816
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A380665.

Programs

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

Formula

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

A380674 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x * (1 - x)^2) ).

Original entry on oeis.org

1, 3, 25, 370, 8097, 237096, 8733601, 388380000, 20253654945, 1212334652800, 81937521020841, 6172429566120192, 512850795552978625, 46594245206418954240, 4595466275857015549425, 488993161791784338804736, 55839856392986843905585089, 6811561624203525171739852800
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377832, A380665, A380666, A380675.

Programs

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

Formula

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

A380780 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * (1 + x)) / (1 + x)^2 ).

Original entry on oeis.org

1, 3, 27, 436, 10353, 326856, 12920731, 614694816, 34223383809, 2184028353280, 157223422977531, 12606338448248832, 1114292924502666673, 107657947282494206976, 11287975339133863810875, 1276603658863119005618176, 154909721707963344338403969, 20076669149268201122957819904
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Links

Crossrefs

Cf. A377829, A380778, A380779, A380781.
Cf. A380665.

Programs

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

Formula

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

A380675 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x * (1 - x)) ).

Original entry on oeis.org

1, 3, 27, 436, 10377, 329016, 13079971, 626414496, 35132554449, 2259697340800, 164013549475371, 13263204195136512, 1182645846100592473, 115285805003164594176, 12197859187688440506675, 1392237638583170475298816, 170517388925776876433310369, 22307473046095249063001554944
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377832, A380665, A380666, A380674.

Programs

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

Formula

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

A380721 E.g.f. A(x) satisfies A(x) = exp(x / (1 - x*A(x))^3) / (1 - x*A(x))^2.

Original entry on oeis.org

1, 3, 29, 511, 13313, 462401, 20140495, 1056765711, 64931273601, 4575023966017, 363744086548751, 32219262817769039, 3146690718151835233, 335963164545043929921, 38931639595489583488239, 4866587415704561667715471, 652773358729046023136421377, 93523037570967777721191018881
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Cf. A380665, A380722.

Programs

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

Formula

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

A380722 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 * (1 - x*A(x))) / (1 - x*A(x))^2.

Original entry on oeis.org

1, 3, 33, 679, 20905, 863601, 44912347, 2820755183, 207815625073, 17578781394913, 1679410405425571, 178871724214036767, 21017369600310686665, 2700840226820242034321, 376826763817725194699083, 56730569139675562422229711, 9166624006966363722766482913, 1582356756863532248954506939329
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Cf. A380665, A380721.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (n+k+1)^(k-1) * binomial(3*n+1,n-k)/k!.
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.