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 1-2 of 2 results.

A363311 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^3 + A(x)^5).

Original entry on oeis.org

1, 2, 16, 180, 2360, 33760, 510928, 8043440, 130371936, 2161066432, 36465401344, 624274702464, 10816259970048, 189305983870208, 3341924242051840, 59437975940616960, 1064030847809734144, 19157066319365860352, 346663014660754833408, 6301645517153393121280
Offset: 0

Views

Author

Paul D. Hanna, May 29 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 180*x^3 + 2360*x^4 + 33760*x^5 + 510928*x^6 + 8043440*x^7 + 130371936*x^8 + 2161066432*x^9 + 36465401344*x^10 + ...
where A(x) = 1 + x*(A(x)^3 + A(x)^5).
RELATED SERIES.
A(x)^3 = 1 + 6*x + 60*x^2 + 740*x^3 + 10200*x^4 + 150576*x^5 + 2328640*x^6 + 37242096*x^7 + ...
A(x)^5 = 1 + 10*x + 120*x^2 + 1620*x^3 + 23560*x^4 + 360352*x^5 + 5714800*x^6 + 93129840*x^7 + ... + A363310(n-1)*x^n + ...

Links

Crossrefs

Cf. A363309, A363310, A027307, A001764.

Programs

  • Maple
    A363311 := proc(n)
    add(binomial(n,k)*binomial(3*n+2*k+1,n)/(3*n+2*k+1),k=0..n) ;
end proc:
seq(A363311(n),n=0..70) ; # R. J. Mathar, Jul 18 2023
  • PARI
    {a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n+2*k+1, n)/(3*n+2*k+1) )}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = 1 + x*(A(x)^3 + A(x)^5).
(2) A(x) = ((B(x) - 1)/x)^(1/5) where B(x) is the g.f. of A363310.
(3) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(3*n+2*k+1, n)/(3*n+2*k+1) for n >= 0.
D-finite with recurrence +8*n*(9639909229907389*n -4332180801077160)* (4*n+1) *(2*n-1) *(4*n-1) *a(n) +(-76286895522125418545*n^5 +381775644252842912682*n^4 -1033993649015194853931*n^3 +1551245138730960078498*n^2 -1139936487176542639744*n +315922393907140666080) *a(n-1) +2*(272671960126472445261*n^5 -3010900995907383509536*n^4 +12907236726784549786263*n^3 -27012522362058892089464*n^2 +27708850835094249342996*n -11174516509692301247280) *a(n-2) +4*(-627566489435411923*n^5 +144061968293307107646*n^4 -1706290600068411299693*n^3 +7720188970563268791354*n^2 -15561118085635458987024*n +11755034318370549299520) *a(n-3) -8*(n-4) *(696748847001815555*n^4 -19100265029551686306*n^3 +142472091583377235329*n^2 -415309555491080054458*n +422902881832258952040) *a(n-4) -96*(n-4) *(n-5)*(3*n-13) *(2465432947213573*n -7363340799047272) *(3*n-14) *a(n-5)=0. - R. J. Mathar, Jul 18 2023
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A363309 Expansion of g.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 8, 67, 590, 5403, 51034, 494268, 4886794, 49153835, 501631980, 5182767291, 54115252508, 570206217940, 6055948422280, 64765311313944, 696876526961130, 7539151412082315, 81957518070961472, 894826829565106185, 9808173152466891270, 107888887505651377475
Offset: 0

Views

Author

Paul D. Hanna, May 29 2023

Keywords

Comments

Compare the g.f. A(x) = F(x*F(x)^5) to F(-x*F(x)^5) = 1/F(x), where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
Conjecture: given A(x) = F(x*F(x)^(2*n-1)) where F(x) = 1 + x*F(x)^n, let B(x) = A(x*B(x)^(n-1)), then ((B(x) - 1)/x)^(1/(2*n-1)) is an integer series for n >= 1. Incidentally, the function A(x) = F(x*F(x)^(2*n-1)) is interesting because F(-x*F(x)^(2*n-1)) = 1/F(x) when F(x) = 1 + x*F(x)^n. This sequence illustrates the case for n = 3; for n = 2, see A363308.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 67*x^3 + 590*x^4 + 5403*x^5 + 51034*x^6 + 494268*x^7 + 4886794*x^8 + 49153835*x^9 + 501631980*x^10 + ...
such that A(x) = F(x*F(x)^5) where F(x) = 1 + x*F(x)^3 begins
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + ... + A001764(n)*x^n + ...
RELATED SERIES.
Let B(x) = A(x*B(x)^2) which begins
B(x) = 1 + x + 10*x^2 + 120*x^3 + 1620*x^4 + 23560*x^5 + 360352*x^6 + 5714800*x^7 + 93129840*x^8 + ... + A363310(n)*x^n + ...
then
( (B(x) - 1)/x )^(1/5) = 1 + 2*x + 16*x^2 + 180*x^3 + 2360*x^4 + 33760*x^5 + 510928*x^6 + 8043440*x^7 + ... + A363311(n)*x^n + ...
is an integer series.

Links

Crossrefs

Cf. A363308, A363309, A363310, A363311, A363111, A001764.

Programs

  • PARI
    {a(n) = if(n==0, 1, sum(k=1, n, 5*k* binomial(3*k+1, k) * binomial(3*n+2*k, n-k) / ((3*k+1)*(3*n+2*k)) ) )}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* G.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 */
{a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^3 + x*O(x^n));
polcoeff( subst(F, x, x*F^5), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, F(x) is the g.f. of A001764.
(1) A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3.
(2) A(x) = B(x/A(x)^2) where B(x) = A(x*B(x)^2) = F( x*B(x)^2 * F(x*B(x)^2)^5 ) is the g.f. of A363310.
(3) a(n) = Sum_{k=1..n} 5*k* binomial(3*k+1, k) * binomial(3*n+2*k, n-k) / ((3*k+1)*(3*n+2*k)) for n > 0, with a(0) = 1.
Showing 1-2 of 2 results.

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