A363311 -id:A363311 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, May 29 2023

nonn

			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 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A363309, A363311, A001764.

{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), ", "))

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.

A363304 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^4 + A(x)^7).

Original entry on oeis.org

1, 2, 22, 350, 6538, 133658, 2895214, 65294502, 1516963346, 36056007602, 872615973766, 21430572885422, 532737957899290, 13379121740808266, 338941379999841758, 8651415618928816886, 222278432539991439906, 5743974149517874477922, 149192980850883703986166

Offset: 0

Paul D. Hanna, May 29 2023

nonn

			G.f.: A(x) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + 1516963346*x^8 + 36056007602*x^9 + ...
where A(x) = 1 + x*(A(x)^4 + A(x)^7).
RELATED SERIES.
A(x)^4 = 1 + 8*x + 112*x^2 + 1960*x^3 + 38528*x^4 + 813064*x^5 + 17998512*x^6 + 412364968*x^7 + ...
A(x)^7 = 1 + 14*x + 238*x^2 + 4578*x^3 + 95130*x^4 + 2082150*x^5 + 47295990*x^6 + 1104598378*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A027307, A363311, A363111.

{a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+3*k+1, n)/(4*n+3*k+1) )}
for(n=0, 20, print1(a(n), ", "))

G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = 1 + x*(A(x)^4 + A(x)^7).
(2) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(4*n+3*k+1, n)/(4*n+3*k+1) for n >= 0.

A363309 Expansion of g.f. A(x) = F(xF(x)^5), where F(x) = 1 + xF(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

Paul D. Hanna, May 29 2023

nonn

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.

			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.

Paul D. Hanna, Table of n, a(n) for n = 0..300

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

{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), ", "))

/* 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), ", "))

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.

A364167 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^3 * (1 + A(x)^3).

Original entry on oeis.org

1, 2, 18, 234, 3570, 59586, 1053570, 19392490, 367677090, 7131417282, 140834140722, 2822214963882, 57243994984722, 1172991472484610, 24245748916730658, 504935751379031082, 10584721220759172162, 223163804001804187266, 4729176407109705542994, 100676187744957784842090

Offset: 0

Seiichi Manyama, Jul 13 2023

Alois P. Heinz, Table of n, a(n) for n = 0..737 (first 50 terms from Christian N. Hofmann)

Cf. A144097, A153231, A363311.

a:= n-> sum(binomial(n, k)*binomial(3*n+3*k+1, n)/(3*n+3*k+1), k=0..n):
seq(a(n), n=0..49); # Christian N. Hofmann, Jul 14 2023

a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n+3*k+1, n)/(3*n+3*k+1));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n+3*k+1,n)/(3*n+3*k+1).

A364397 G.f. satisfies A(x) = 1 + x/A(x)^2*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -12, 124, -1560, 21776, -324256, 5046096, -81086112, 1335113408, -22408067200, 381942129792, -6593494698752, 115044039049728, -2025580621035520, 35943759448886528, -642162301086308864, 11541259115333684224, -208521418711421405184

Offset: 0

Seiichi Manyama, Jul 22 2023

sign

Cf. A364393, A364395, A364399.

Cf. A363311.

a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(3*n+2*k-2, n-1))/n);

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A363311.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(3*n+2*k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n+1)*27^n*4^(-n)*3F2([-n, 3*n/2, (3n-1)/2], [n, n+1/2], -1)*n^(-3/2), with c = 1/(3*sqrt(3*Pi)). - Stefano Spezia, Oct 21 2023

A371661 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x))^2.

Original entry on oeis.org

1, 4, 64, 1424, 36800, 1036160, 30843648, 954671360, 30415326208, 990831196160, 32853724512256, 1105132250898432, 37620337933582336, 1293586791397064704, 44863864476704768000, 1567543145774827241472, 55125711913212153954304

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..633

Cf. A271469, A363311, A371660.

Cf. A371669.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(4*n-k, n-1-2*k))/n);

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 4^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0.

a(n) = 2^n * A371669(n). - Seiichi Manyama, Dec 26 2024

A363305 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^5 + A(x)^9).

Original entry on oeis.org

1, 2, 28, 576, 13968, 371280, 10465152, 307252032, 9295409664, 287758274304, 9071667965184, 290237226038272, 9399819302979584, 307570021821937664, 10152439243763290112, 337658352835320934400, 11304320019217804476416, 380650592731460987617280

Offset: 0

Paul D. Hanna, May 29 2023

nonn

			G.f.: A(x) = 1 + 2*x + 28*x^2 + 576*x^3 + 13968*x^4 + 371280*x^5 + 10465152*x^6 + 307252032*x^7 + 9295409664*x^8 + ...
where A(x) = 1 + x*(A(x)^5 + A(x)^9).
RELATED SERIES.
A(x)^5 = 1 + 10*x + 180*x^2 + 4080*x^3 + 104160*x^4 + 2858352*x^5 + 82336320*x^6 + 2455727040*x^7 + ...
A(x)^9 = 1 + 18*x + 396*x^2 + 9888*x^3 + 267120*x^4 + 7606800*x^5 + 224915712*x^6 + 6839682624*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A027307, A363311, A363304.

{a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+4*k+1, n)/(5*n+4*k+1) )}
for(n=0, 20, print1(a(n), ", "))

G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = 1 + x*(A(x)^5 + A(x)^9).
(2) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(5*n+4*k+1, n)/(5*n+4*k+1) for n >= 0.

A363380 G.f. satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^2).

Original entry on oeis.org

1, 2, 20, 284, 4712, 85392, 1638112, 32699472, 672188768, 14133399744, 302535052160, 6570819330688, 144442463464704, 3207564324825600, 71848240540852224, 1621452789508328704, 36831997860270007808, 841470878382566444032

Offset: 0

Seiichi Manyama, May 29 2023

Cf. A219534, A346626, A363311.

Cf. A217360, A260332, A363304.

a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+2*k+1, n)/(4*n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k+1,n)/(4*n+2*k+1).

A364195 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^5 * (1 + A(x)^2).

Original entry on oeis.org

1, 2, 24, 412, 8280, 181904, 4232048, 102479184, 2555884896, 65207430848, 1693785940992, 44643489969792, 1190986788639232, 32097745138518528, 872595854798515456, 23900545715576753408, 658934625866433496576, 18271554709525993556992, 509241947434834351042560

Offset: 0

Seiichi Manyama, Jul 13 2023

Cf. A217364, A363006, A363305, A364196.

Cf. A349310, A363304, A363311.

a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+2*k+1, n)/(5*n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+2*k+1,n)/(5*n+2*k+1).

A371660 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x) + A(x)^2).

Original entry on oeis.org

1, 3, 36, 603, 11745, 249372, 5599044, 130735620, 3142426428, 77238209502, 1932396279066, 49047725266101, 1259884849971465, 32690034127387431, 855528520866461010, 22556952666651901761, 598607836414445357145, 15976563963437863357146

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..685

Cf. A271469, A363311, A371661.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 3^(n-k)*binomial(n, k)*binomial(4*n-k, n-1-2*k))/n);

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 3^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0.

cp's OEIS Frontend

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.

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A363304 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^4 + A(x)^7).

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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.

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A364167 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^3 * (1 + A(x)^3).

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A364397 G.f. satisfies A(x) = 1 + x/A(x)^2*(1 + 1/A(x)^2).

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A371661 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x))^2.

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A363305 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^5 + A(x)^9).

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A363380 G.f. satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^2).

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A364195 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^5 * (1 + A(x)^2).

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

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