A380514 -id:A380514 - cp's OEIS frontend

A251568 Expansion of e.g.f. exp(xC(x)^2) where C(x) = 1 + xC(x)^2 is the g.f. of the Catalan numbers, A000108.

1, 1, 5, 43, 529, 8501, 169021, 4010455, 110676833, 3484717129, 123320412181, 4847038223171, 209536628422705, 9882471447634813, 505033804901100749, 27802319803528367791, 1640388588050579832001, 103275015543414629215505, 6910877628962983581031333

Offset: 0

Paul D. Hanna, Dec 05 2014

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 529*x^4/4! + 8501*x^5/5! + ...
where
log(A(x)) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ... + A000108(n)*x^n + ...

G. C. Greubel, Table of n, a(n) for n = 0..360

Cf. A080893, A250917.
Cf. A082579, A380511, A380514.
Cf. A001517, A251569, A000108, A382032.

CatalanNumber := n -> binomial(2*n,n)/(n+1):
a := n -> `if`(n=0, 1, n!*CatalanNumber(n)*hypergeom([1-n], [2+n], -1)):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 04 2017

Flatten[{1,Table[Sum[n!/k!*Binomial[2*n-1,n-k]*2*k/(n+k),{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 14 2015 *)
a[0] = 1; a[n_] := (2n)!/(n+1)! Hypergeometric1F1[1-n, n+2, -1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 03 2017, after Vladimir Kruchinin *)

{a(n)=my(C=1);for(i=1,n,C=1+x*C^2 +x*O(x^n));n!*polcoef(exp(x*C^2),n)}
for(n=0,20,print1(a(n),", "))

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(serreverse(x*(1-x))^2/x))) \\ Seiichi Manyama, Mar 15 2025

a(n) = Sum_{k=0..n} (n!/k!) * binomial(2*n-1, n-k) * 2*k/(n+k) for n > 0 with a(0)=1.
E.g.f. A(x) satisfies: A'(x)/A(x) = C'(x) = C(x)^2 / sqrt(1-4*x) where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function.
Recurrence equation: a(n) = -(n^2 - 5*n +1)*a(n-1) + n*(2*n - 3)*(2*n - 4)*a(n-2) with a(0) = 1, a(1) = 1. It appears that a(n) - 1 is divisible by n*(n - 1) for n >= 2. - Peter Bala, Feb 14 2015
a(n) ~ 2^(2*n+1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Feb 14 2015
a(n) are special values of the hypergeometric function 1F1: a(n) = 4^n*Gamma(n+1/2)*exp(-1)*hypergeom([2*n+1], [n+2], 1)/(sqrt(Pi)*(n+1)), for n>=1. - Karol A. Penson, Jun 01 2015
a(n) = ((2*n)!/(n+1)!)*hypergeometric([1-n],[n+2],-1), a(0)=1. - Vladimir Kruchinin, May 03 2017
From Seiichi Manyama, Mar 15 2025: (Start)
E.g.f.: exp( (1/x) * Series_Reversion( x*(1-x) )^2 ).
E.g.f.: exp( Series_Reversion( x/(1+x)^2 ) ). (End)

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

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

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A380516 Expansion of e.g.f. exp(xG(x)^4) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 157, 4129, 146001, 6502681, 349790029, 22069858497, 1598577634369, 130757736096361, 11922399644742621, 1199121973234651489, 131887738425602277457, 15748194681225620534649, 2028885239529647188594381, 280525944581514367875035521, 41434950383158772951280658689

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A002293, A380513, A380514, A380515.
Cf. A000262, A251568, A380512.
Cf. A382034.

Join[{1}, Table[(n-1)! * LaguerreL[n-1, 3*n+1, -1], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 26 2025 *)

a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 3*n+1, -1));

E.g.f.: exp(G(x)-1), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} binomial(4*n,k)/(n-k-1)! for n > 0.
a(n+1) = n! * LaguerreL(n, 3*n+4, -1).
a(n) = (-1)^(n+1)*U(1-n, 2+3*n, -1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
a(n) ~ 2^(8*n + 1) * n^(n-1) / (exp(n - 1/3) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jan 26 2025
E.g.f.: exp( Series_Reversion( x/(1+x)^4 ) ). - Seiichi Manyama, Mar 15 2025

A380511 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 55, 961, 23141, 711421, 26631235, 1175535425, 59786520841, 3442729157461, 221413508687471, 15730688410899265, 1223574846548300845, 103417508018836074701, 9437941200860641295611, 924934291227615821904001, 96881241931552168636182545, 10801002623361396194857667365

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A251569, A380512.
Cf. A080893, A380515.
Cf. A082579, A251568, A380514.
Cf. A001764, A006013, A370054, A382058.

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

a(n) = 2 * n! * Sum_{k=0..n-1} binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-3*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^2 ) ). - Seiichi Manyama, Mar 15 2025

A380513 Expansion of e.g.f. exp(xG(x)) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 31, 649, 20241, 831691, 42281023, 2558247441, 179401012129, 14301145772371, 1276863732880671, 126200478678828313, 13677209933635675441, 1612657716714084149019, 205505541279096688937791, 28144314031348292162103841, 4122178445898981809990411073, 642961375302043479923591655331

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A002293, A380514, A380515, A380516.

Cf. A000262, A080893, A251569.

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

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

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A251568 Expansion of e.g.f. exp(x*C(x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.

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A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A380516 Expansion of e.g.f. exp(x*G(x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A380511 Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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A380513 Expansion of e.g.f. exp(x*G(x)) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A251568 Expansion of e.g.f. exp(xC(x)^2) where C(x) = 1 + xC(x)^2 is the g.f. of the Catalan numbers, A000108.

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

A380516 Expansion of e.g.f. exp(xG(x)^4) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

A380511 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

A380513 Expansion of e.g.f. exp(xG(x)) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.