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 1-6 of 6 results.

A377553 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^2 ).

Original entry on oeis.org

1, 2, 14, 174, 3176, 77010, 2336892, 85316714, 3644408336, 178412603778, 9851421767060, 605826315779322, 41068369222584024, 3042849619010389058, 244657525386435161756, 21217387476442659806250, 1974219906922046702054432, 196191093901292764305110274, 20739322455031604846405387556
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2024

Keywords

Links

Crossrefs

Cf. A161633, A377554.
Cf. A364982.

Programs

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

Formula

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364982.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(2*n+2,k)/(n-k)!.

A382033 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 109, 2653, 88261, 3731581, 191571493, 11576241769, 804996352873, 63324553740121, 5559962513556001, 539015912053933645, 57188111522488589293, 6591136171961660099509, 820029701725988751533341, 109537705061927547203868241, 15635869913619342121140932689
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A161630, A382032, A382034.
Cf. A001764, A377554.

Programs

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

Formula

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377554. F(x) = log(A(x))/x = B(x*A(x))^3.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^3 ) ).

A377575 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 30, 483, 11100, 334035, 12478698, 558058179, 29104042152, 1735547479587, 116539815603630, 8704631976941043, 716019297815418732, 64326542671867079955, 6267631435921525638738, 658359915933162131600355, 74168964857766293453918928, 8921104769819780822122624323
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377574, A377576.
Cf. A364983, A377554.

Programs

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

Formula

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

A381443 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sinh(x))^3 ).

Original entry on oeis.org

1, 3, 24, 333, 6720, 179523, 5992800, 240498261, 11287790592, 607019415075, 36813049552896, 2486167829854173, 185070328813031424, 15056826823777670883, 1329283990371617820672, 126573877370649849898149, 12930948581449447912243200, 1410875453109072905123881923
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Links

Crossrefs

Cf. A377554, A381450.
Cf. A136630, A381430.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*n+3, k)*a136630(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + sinh(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381430.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(3*n+3,k) * A136630(n,k).

A381450 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^3 ).

Original entry on oeis.org

1, 3, 24, 339, 7056, 195855, 6819840, 286105071, 14055420288, 791783681499, 50327779368960, 3563709848656683, 278223968271034368, 23744747385054558759, 2199369837961901789184, 219748696455778150645575, 23559108001707680103628800, 2697737574531326391439989171
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A381171, A381449.
Cf. A377554, A381443.
Cf. A185951, A381448.

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(3*n+3, k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cosh(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381448.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(3*n+3,k) * A185951(n,k).
Showing 1-6 of 6 results.

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