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.

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

A363308 Expansion of g.f. C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 5, 26, 141, 790, 4542, 26668, 159333, 966038, 5930678, 36801660, 230491410, 1455283172, 9253674120, 59209786992, 380961295445, 2463303690790, 15998687418030, 104325569140156, 682768883525830, 4483232450501492, 29527005540912660, 195006621974036808
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

Compare the g.f. A(x) = C(x*C(x)^3) to the identity C(-x*C(x)^3) = 1/C(x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
Conjecture: a(n) is odd iff n is a power of 2 or n = 0.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 141*x^4 + 790*x^5 + 4542*x^6 + 26668*x^7 + 159333*x^8 + 966038*x^9 + 5930678*x^10 + ...
such that A(x) = C(x*C(x)^3), where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... + A000108(n)*x^n + ...
x*C(x)^3 = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + ... + A000245(n)*x^n + ...
Note that x*C(x)^3 = (C(x) - 1)*(1-x)/x - 1.
Also, the g.f. of related sequence A033296 begins
B(x) = 1 + x + 6*x^2 + 42*x^3 + 326*x^4 + 2706*x^5 + 23526*x^6 + ...
where A(x) = B(x/A(x)), B(x) = A(x*B(x)) = C(x*B(x)*C(x*B(x))^3).

Links

Crossrefs

Cf. A127632, A153294, A033296, A000108 (C(x)), A000245 (x*C(x)^3).
Cf. A363309, A363111.

Programs

  • PARI
    {a(n) = if(n==0,1, sum(k=1,n, 3*k* binomial(2*k+1,k) * binomial(2*n+k,n-k) / ((2*k+1)*(2*n+k)) ) )}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* G.f. A(x) = C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 */
{a(n) = my(C = (1 - sqrt(1 - 4*x +x^2*O(x^n)))/(2*x)); polcoeff( subst(C, x, x*C^3), 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, C(x) is the g.f. of the Catalan numbers (A000108).
(1) A(x) = C(x*C(x)^3), where C(x) = (1 - sqrt(1-4*x))/(2*x).
(2) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = C( x*B(x) * C(x*B(x))^3 ) is the g.f. of A033296.
(3) a(n) = Sum_{k=1..n} 3*k* binomial(2*k+1,k) * binomial(2*n+k,n-k) / ((2*k+1)*(2*n+k)) for n > 0, with a(0) = 1.
D-finite with recurrence 4*n*(n-1)*(21687905*n +56141583)*(n+1)*a(n) +2*n*(n-1) *(43375810*n^2 -6022811713*n +10976463649)*a(n-1) -(n-1) *(15429963345*n^3 -200018809315*n^2 +658353214412*n -632905646028)*a(n-2) +(102558230760*n^4 -1409936457473*n^3 +6909548744112*n^2 -14414518702669*n +10812683474490)*a(n-3) +(-212869593020*n^4 +3377685007909*n^3 -20069314453381*n^2 +52902205420466*n -52146873039204)*a(n-4) +4*(2*n-11) *(10773532140*n^3 -171459615587*n^2 +902576783797*n -1572525214995)*a(n-5) +4*(n-6) *(208500820*n -955419151)*(2*n-11) *(2*n-13)*a(n-6)=0. - R. J. Mathar, Nov 22 2024

A363111 Expansion of g.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 11, 127, 1547, 19652, 258069, 3481034, 47999915, 674086924, 9612919156, 138878011335, 2028718584989, 29918897595468, 444889269572286, 6663228661354420, 100430376524360459, 1522215623202615036, 23187346871707554564, 354783440893854307244
Offset: 0

Views

Author

Paul D. Hanna, May 30 2023

Keywords

Comments

Compare the g.f. A(x) = F(x*F(x)^7) to F(-x*F(x)^7) = 1/F(x), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

Examples

			G.f.: A(x) = 1 + x + 11*x^2 + 127*x^3 + 1547*x^4 + 19652*x^5 + 258069*x^6 + 3481034*x^7 + 47999915*x^8 + 674086924*x^9 + ...
such that A(x) = F(x*F(x)^7) where F(x) = 1 + x*F(x)^4 begins
F(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 + 53820*x^7 + ... + A002293(n)*x^n + ...
RELATED SERIES.
Let B(x) = A(x*B(x)^3) = ( Series_Reversion( x/A(x)^3 )/x )^(1/3) which begins
B(x) = 1 + x + 14*x^2 + 238*x^3 + 4578*x^4 + 95130*x^5 + 2082150*x^6 + 47295990*x^7 + 1104598378*x^8 + ...
then
( (B(x) - 1)/x )^(1/7) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + ... + A363304(n)*x^n + ...
is an integer series.

Crossrefs

Cf. A363304, A363308, A363309, A002293.

Programs

  • PARI
    {a(n) = if(n==0, 1, sum(k=1, n, 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) ) )}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* G.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 */
{a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^4 + x*O(x^n));
polcoeff( subst(F, x, x*F^7), 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 A002293.
(1) A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4.
(2) A(x) = B(x/A(x)^3) where B(x) = A(x*B(x)^3) = F( x*B(x)^3 * F(x*B(x)^3)^7 ).
(3) a(n) = Sum_{k=1..n} 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) for n > 0, with a(0) = 1.

A363310 Expansion of g.f. A(x) satisfying A(x) = 1 + x*G(x)^5, where G(x) = 1 + x*(G(x)^3 + G(x)^5) is the g.f. of A363311.

Original entry on oeis.org

1, 1, 10, 120, 1620, 23560, 360352, 5714800, 93129840, 1550132320, 26242225600, 450448137216, 7821608426880, 137145465358080, 2424899712359680, 43186456105340160, 774013543036174080, 13949937641606981120, 252666943472167541760, 4596736161565468815360
Offset: 0

Views

Author

Paul D. Hanna, May 29 2023

Keywords

Examples

			G.f.: A(x) =  = 1 + x + 10*x^2 + 120*x^3 + 1620*x^4 + 23560*x^5 + 360352*x^6 + 5714800*x^7 + 93129840*x^8 + 1550132320*x^9 + 26242225600*x^10 + ...
such that A(x) = 1 + x*G(x)^5 where
G(x) = 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 + ...
satisfies G(x) = 1 + x*(G(x)^3 + G(x)^5).
Also, A(x) = B(x*A(x)^2) where B(x) = A(x/B(x)^2) begins
B(x) = 1 + x + 8*x^2 + 67*x^3 + 590*x^4 + 5403*x^5 + 51034*x^6 + 494268*x^7 + ... + A363309(n)*x^n + ...

Links

Crossrefs

Cf. A363309, A363311, A001764.

Programs

  • PARI
    {a(n) = if(n==0,1, sum(k=0, n-1, 5*binomial(n-1, k)*binomial(3*n+2*k+2, n-1)/(3*n+2*k+2) ) )}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, B(x) is the g.f. of A363309 and F(x) is the g.f. of A001764.
(1) A(x) = 1 + x*G(x)^5, where G(x) = 1 + x*(G(x)^3 + G(x)^5) is the g.f. of A363311.
(2) A(x) = B(x*A(x)^2) where B(x) = F(x*F(x)^5) and F(x) = 1 + x*F(x)^3.
(3) A(x) = sqrt( (1/x)*Series_Reversion( x/B(x)^2 ) ), where B(x) is the g.f. of A363309.
(4) a(n) = Sum_{k=0..n-1} 5*binomial(n-1, k)*binomial(3*n+2*k+2, n-1)/(3*n+2*k+2) for n > 1 with a(0) = 1.
Showing 1-4 of 4 results.

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