A382032 -id:A382032 - 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)

A382033 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 109, 2653, 88261, 3731581, 191571493, 11576241769, 804996352873, 63324553740121, 5559962513556001, 539015912053933645, 57188111522488589293, 6591136171961660099509, 820029701725988751533341, 109537705061927547203868241, 15635869913619342121140932689

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161630, A382032, A382034.

Cf. A001764, A377554.

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

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377554. F(x) = log(A(x))/x = B(x*A(x))^3.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^3 ) ).

A382034 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 181, 5713, 246881, 13570081, 906180997, 71250724833, 6448375469665, 660286026034561, 75472025139452261, 9525947428687403473, 1315935073971181422721, 197485196722573989608289, 31993978774204625549549221, 5565216938342017912128576961, 1034506012356981473110554574145

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161630, A382032, A382033.

Cf. A002293, A377630.

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

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(4*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377630. F(x) = log(A(x))/x = B(x*A(x))^4.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^4 ) ).

A382036 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)^2) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 7, 94, 1901, 51696, 1771267, 73317616, 3560476761, 198531343360, 12502959204671, 877829600807424, 67991178144166213, 5759309535250776064, 529665762441463234875, 52560256640090731902976, 5597859153748148214250673, 636915477940535101583130624, 77102760978489789146276986231

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A052873, A382037, A382038.

Cf. A000108, A377829, A382032.

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

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))^2).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^2 ) ).
a(n) ~ 2^(n - 1/2) * n^(n-1) / ((sqrt(2) - 1)^(n - 1/2) * exp((sqrt(2) - 1)*(sqrt(2)*n - 1))). - Vaclav Kotesovec, Mar 15 2025

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 67, 1465, 44541, 1735681, 82527439, 4632741905, 299875704697, 21989097804961, 1801520077445331, 163092373817762137, 16168084561101716725, 1741946677697976052577, 202668693570279026375671, 25324088113475137179021601, 3382305512670022948599733233, 480858973986045019386825360577

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382059.
Cf. A161635, A382032.
Cf. A001764, A377546, A382033.

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

Let F(x) be the e.g.f. of A377546. F(x) = log(A(x))/x = B(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^2 ) ).
a(n) = 2 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n+k,k)/((2*n+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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A382033 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A382034 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A382036 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)^2) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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A382058 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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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.

A382033 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

A382034 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.