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.

A274735 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x*A(x)^2).

Original entry on oeis.org

1, 4, 26, 210, 1901, 18445, 187524, 1971672, 21263360, 233907762, 2614446624, 29607343948, 338977591904, 3917185497535, 45629006313280, 535199773167207, 6315789123860388, 74932400322972992, 893276792585933870, 10694510040508714014, 128531711285410216883, 1550159476645634696615, 18755239991772817629972, 227577929298568261967650, 2768820313297861609739979
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2016

Keywords

Comments

More generally, if G(x) satisfies
G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then
G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 210*x^3 + 1901*x^4 + 18445*x^5 + 187524*x^6 + 1971672*x^7 + 21263360*x^8 +...

Links

Crossrefs

Cf. A007863, A274734.
Cf. A274379, A369600.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = (1 + x*A)^3 * (1 + x*A^2) + x*O(x^n) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x*(1+x)^3)/(1+x +x^2*O(x^n) )^3 ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^3) / (1+x)^3 ).
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(3*n+3*k+3,n-k). - Seiichi Manyama, Jan 27 2024

A369616 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x) ).

Original entry on oeis.org

1, 3, 12, 58, 314, 1824, 11107, 69955, 451918, 2977834, 19936332, 135225006, 927267595, 6417580459, 44770275705, 314489676679, 2222549047262, 15791353483602, 112734135824404, 808247711066688, 5817056710700424, 42012120642574732, 304384379305912686
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2024

Keywords

Links

Crossrefs

Cf. A007317, A369617, A370844.
Cf. A006013, A274734.

Programs

  • Maple
    A369616 := proc(n)
    add(binomial(n+1,k) * binomial(3*n-3*k+1,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369616(n),n=0..70) ; # R. J. Mathar, Jan 28 2024
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^2+x))/x)
  • PARI
    a(n) = sum(k=0, n, binomial(n+1, k)*binomial(3*n-3*k+1, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(3*n-3*k+1,n-k).
D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) +3*(-13*n^2+1)*a(n-1) +33*(2*n-1)*(n-1)*a(n-2) -31*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jan 28 2024

A369599 Expansion of (1/x) * Series_Reversion( x * (1/(1+x)^2 - x^3) ).

Original entry on oeis.org

1, 2, 5, 15, 54, 223, 993, 4580, 21521, 102563, 495318, 2422302, 11979965, 59824535, 301202673, 1527118720, 7789673832, 39947163395, 205835776301, 1065155017623, 5533253267649, 28844759080896, 150846487065730, 791163319140664, 4160593763997122
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2024

Keywords

Links

Crossrefs

Cf. A198888, A369600.
Cf. A274378, A274734.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(2*n+2*k+2,n-3*k).
Showing 1-3 of 3 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.