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.

A367789 E.g.f. satisfies A(x) = exp( x/(1-x)^3 * A(x) ).

Original entry on oeis.org

1, 1, 9, 106, 1697, 35076, 893947, 27165706, 960298593, 38751082552, 1758831242291, 88726543365054, 4926355857050641, 298605321687360676, 19623211558172733435, 1389870724939251455506, 105556814502357807727553, 8557797733469700008170224
Offset: 0

Views

Author

Seiichi Manyama, Nov 30 2023

Keywords

Links

Crossrefs

Cf. A052868, A362775, A367790.
Cf. A091695, A360100.

Programs

  • Maple
    A367789 := proc(n)
    n!*add((k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!,k=0..n) ;
end proc:
seq(A367789(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))))

Formula

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

A360102 a(n) = Sum_{k=0..n} binomial(n+2*k,n-k) * Catalan(k).

Original entry on oeis.org

1, 2, 7, 30, 141, 703, 3655, 19603, 107679, 602756, 3426049, 19721069, 114728723, 673494466, 3984493735, 23732956453, 142204128507, 856560123504, 5183708936061, 31502904805922, 192180259402691, 1176416604202925, 7223943302003917, 44486888142708088
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums of A360100.
Partial sums are A258973.
Cf. A000108, A162481.
Cf. A006318, A007317, A162476, A360103.

Programs

  • Maple
    A360102 := proc(n)
    add(binomial(n+2*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360102(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^3))))

Formula

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^2.
G.f.: (1/(1-x)) * c(x/(1-x)^3), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +10*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A360101 a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 7, 40, 234, 1432, 9078, 59113, 393125, 2659233, 18240801, 126588424, 887221916, 6271153060, 44652824248, 319990906290, 2306133322704, 16703784324239, 121534039921585, 887845073567240, 6509750423778460, 47888814944642434, 353362258550740732
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums are A360103.
Cf. A000108, A360057.
Cf. A002212, A006319, A162475, A360100.

Programs

  • Maple
    A360101 := proc(n)
    add(binomial(n+4*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360101(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    m = 23;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^5 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k-1, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^5)))

Formula

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^5.
G.f.: c(x/(1-x)^5), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +(-10*n+7)*a(n-1) +(19*n-56)*a(n-2) +10*(-2*n+9)*a(n-3) +5*(3*n-19)*a(n-4) +(-6*n+49)*a(n-5) +(n-10)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

A382921 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) / (1-x)^3 )^3.

Original entry on oeis.org

1, 3, 24, 199, 1776, 16713, 163429, 1644852, 16929576, 177384877, 1885842105, 20292695751, 220595817213, 2418988309494, 26726104358958, 297226167487469, 3324654200094495, 37379224636055040, 422182501323170275, 4788001977121735326, 54502930562354983641
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2025

Keywords

Crossrefs

Cf. A360100, A382919.
Cf. A382615, A382917.

Programs

  • PARI
    a(n, r=3, s=3, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(4/3) / (1-x)^3 )^3.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^3, where B(x) is the g.f. of A382917.

A367281 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 5, 32, 237, 1906, 16179, 142665, 1294115, 11998349, 113194205, 1083131419, 10486939473, 102548233212, 1011333385507, 10047289999536, 100458873883179, 1010138430187185, 10208244014494347, 103625607305637693, 1056166710786300973
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2023

Keywords

Crossrefs

Cf. A002294, A360100, A365150, A367240.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).

A376159 G.f. satisfies A(x) = 1 / ((1-x)^3 - x*A(x)).

Original entry on oeis.org

1, 4, 17, 90, 539, 3451, 23100, 159720, 1131905, 8178326, 60019533, 446166771, 3352530190, 25422458170, 194302002463, 1495223230621, 11575504625874, 90090318248607, 704480581789900, 5532228951823605, 43610427926723780, 344972119634359080, 2737451123900901555
Offset: 0

Views

Author

Seiichi Manyama, Sep 12 2024

Keywords

Crossrefs

Cf. A000108, A006318, A162477.
Cf. A052529, A366184, A376160.
Cf. A360100.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^3+sqrt((1-x)^6-4*x)))
  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k+2, n-k)*binomial(2*k, k)/(k+1));

Formula

G.f.: 2 / ((1-x)^3 + sqrt((1-x)^6 - 4*x)).
a(n) = Sum_{k=0..n} binomial(n+5*k+2,n-k) * binomial(2*k,k)/(k+1).

A382919 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) / (1-x)^3 )^2.

Original entry on oeis.org

1, 2, 13, 84, 580, 4216, 31824, 247168, 1962800, 15866016, 130122304, 1080101760, 9057113472, 76610188544, 652895283200, 5600752756224, 48323092761344, 419068973537792, 3650909105378304, 31937405800724480, 280419948474447872, 2470473454986891264
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2025

Keywords

Crossrefs

Cf. A360100, A382921.
Cf. A213282, A382616.

Programs

  • PARI
    a(n, r=2, s=3, t=3, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(3/2) / (1-x)^3 )^2.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^2, where B(x) is the g.f. of A213282.
Showing 1-7 of 7 results.

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

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