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 1-4 of 4 results.

A360100 a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 5, 23, 111, 562, 2952, 15948, 88076, 495077, 2823293, 16295020, 95007654, 558765743, 3310999269, 19748462718, 118471172054, 714355994997, 4327148812557, 26319195869861, 160677354596769, 984236344800234, 6047526697800992, 37262944840704171
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums are A360102.
Cf. A162481, A258973.
Cf. A002212, A006319, A162475, A360101.
Cf. A000108.

Programs

  • Maple
    A360100 := proc(n)
    add(binomial(n+2*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360100(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    m = 24;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^3 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+2*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)^3)))

Formula

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^3.
G.f.: c(x/(1-x)^3), where c(x) is the g.f. of A000108.
a(n) ~ sqrt(-2 + (35 - 3*sqrt(129))^(1/3) + (35 + 3*sqrt(129))^(1/3)) * (((7 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / (sqrt(2*Pi) * n^(3/2))). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence (n+1)*a(n) +(-8*n+5)*a(n-1) +(10*n-27)*a(n-2) +(-4*n+17)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A162476 Expansion of (1/(1-x))*c(x/(1-x)^4), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 8, 39, 205, 1136, 6548, 38882, 236260, 1462131, 9184413, 58408588, 375330536, 2433325315, 15896742423, 104546968252, 691608993478, 4599024778431, 30724413312953, 206114347293697, 1387917616331135, 9377747277136328
Offset: 0

Views

Author

Paul Barry, Jul 04 2009

Keywords

Comments

Partial sums are A162477. Partial sums of A162475.

Formula

G.f.: 1/(1-x-x/((1-x)^3-x/(1-x-x/((1-x)^3-x/(1-x-x/((1-x)^3-x/(1-... (continued fraction);
a(n) = Sum_{k=0..n} C(n+3k,n-k)*A000108(k).
(n+1)*(2*n-3)*a(n) -(4*n-3)*(4*n-5)*a(n-1) +3*(4*n^2-14*n+11)*a(n-2) +(-8*n^2+40*n-27)*a(n-3) +(2*n-1)*(n-6)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011

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

A367790 E.g.f. satisfies A(x) = exp( x/(1-x)^4 * A(x) ).

Original entry on oeis.org

1, 1, 11, 148, 2669, 62056, 1777927, 60692920, 2408692505, 109074596320, 5553702114731, 314208715035304, 19561795753879909, 1329317730339826384, 97924919301787209647, 7773978186375852940696, 661702605336795904770353, 60119367618216155944350400
Offset: 0

Views

Author

Seiichi Manyama, Nov 30 2023

Keywords

Links

Crossrefs

Cf. A052868, A362775, A367789.
Cf. A162475, A361283.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^4))))

Formula

E.g.f.: exp( -LambertW(-x/(1-x)^4) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+3*k-1,n-k)/k!.
Showing 1-4 of 4 results.

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