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 1-3 of 3 results.

A380095 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*A(x)^2*exp(x*A(x)^2) ).

Original entry on oeis.org

1, 1, 9, 156, 4129, 147880, 6696591, 367141306, 23648581713, 1750754472840, 146492770433095, 13672570280741086, 1408330043282040825, 158697952371711709060, 19420527592823261136519, 2564857285665551372127570, 363619232307437704055993761, 55079007956127598819416831088
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A213644, A380096.
Cf. A380042, A379688.

Programs

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

Formula

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

A380097 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 3*x*exp(x)) ).

Original entry on oeis.org

1, 3, 42, 1089, 42132, 2182335, 142084818, 11159447943, 1027313395944, 108517938075387, 12940759400071710, 1719811206219287643, 252076045285741340700, 40398758175398949144039, 7028240082095865121961514, 1319141702032289451776382975, 265703833060229155917857703888
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A213644, A379688.
Cf. A380096.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*A(x)*exp(x*A(x)) ).
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380096.
a(n) = (1/(n+1)) * Sum_{k=0..n} 3^k * k^(n-k) * (n+k)!/(k! * (n-k)!).
a(n) == 0 (mod 3) for n>0.

A379687 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) * (1 - 2*x*exp(x)) ).

Original entry on oeis.org

1, 1, 11, 158, 3597, 107994, 4082695, 186003166, 9930511577, 608225910290, 42049314628251, 3239451955702038, 275220868466701861, 25565354021529630970, 2577774089234276245391, 280406846696018760511694, 32732658189778781519050161, 4081497882738710247779141922
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A379659, A379688.

Programs

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

Formula

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

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

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