A080893 -id:A080893 - cp's OEIS frontend

A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.

1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753

Offset: 0

N. J. A. Sloane

Numerators of successive convergents to e using continued fraction 1 + 2/(1 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + 1/(22 + 1/26 + ...)))))).

Number of ways to use the elements of {1,...,k}, n <= k <= 2n, once each to form a collection of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006

L. Euler, 1737.
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 0.126, p. 2.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..334 (first 101 terms from T. D. Noe)
P. Bala, A note on the Catalan transform of a sequence.
J. W. L. Glaisher, On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 131.
D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 149.
D. H. Lehmer, Review of various tables by P. Pederson, Math. Comp., 2 (1946), 68-69.
W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968.
J. Riordan, Letter, Jul 06 1978.
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970.
Index entries for related partition-counting sequences
Index entries for sequences related to Bessel functions or polynomials

Essentially the same as A080893.
a(n) = A099022(n)/n!.
Partial sums: A105747.
Replace "lists" with "sets" in comment: A001515.
Cf. A001515, A001518, A002119, A053556, A053557, A243593.

A:= gfun:-rectoproc({a(n) = (4*n-2)*a(n-1) + a(n-2),a(0)=1,a(1)=3},a(n),remember):
map(A, [$0..20]); # Robert Israel, Jul 22 2015
f:=proc(n) option remember; if n = 0 then 1 elif n=1 then 3 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n), n=0..20)]; # N. J. A. Sloane, May 09 2016
seq(simplify(KummerU(-n, -2*n, 1)), n = 0..16); # Peter Luschny, May 10 2022

Table[(2k)! Hypergeometric1F1[-k, -2k, 1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

PARI
```
a(n)=sum(k=0,n,(n+k)!/k!/(n-k)!)
```

A001517 = lambda n: hypergeometric([-n, n+1], [], -1)
[simplify(A001517(n)) for n in (0..16)] # Peter Luschny, Oct 17 2014

a(n) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!) = (e/Pi)^(1/2) K_{n+1/2}(1/2).
D-finite with recurrence a(n) = (4*n-2)*a(n-1) + a(n-2), n >= 2.
a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*A000522(n+k). - Vladeta Jovovic, Sep 30 2006
E.g.f. (for offset 1): exp(x*c(x)), where c(x)=(1-sqrt(1-4*x))/(2*x) (cf. A000108). - Vladimir Kruchinin, Aug 10 2010
G.f.: 1/Q(0), where Q(k) = 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = (1/n!)*Integral_{x>=0} (x*(1 + x))^n*exp(-x) dx. Expansion of exp(x) in powers of y = x*(1 - x): exp(x) = 1 + y + 3*y^2/2! + 19*y^3/3! + 193*y^4/4! + 2721*y^5/5! + .... - Peter Bala, Dec 15 2013
a(n) = exp(1/2) / sqrt(Pi) * BesselK(n+1/2, 1/2). - Vaclav Kotesovec, Mar 15 2014
a(n) ~ 2^(2*n+1/2) * n^n / exp(n-1/2). - Vaclav Kotesovec, Mar 15 2014
a(n) = hypergeom([-n, n+1], [], -1). - Peter Luschny, Oct 17 2014
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * 4^n * hypergeometric1f1(-n; -2*n; 1).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 4*t/(1-t)^2). (End)
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*k!. - Ilya Gutkovskiy, Nov 24 2017
a(n) = KummerU(-n, -2*n, 1). - Peter Luschny, May 10 2022

More terms from Vladeta Jovovic, Apr 03 2000

Additional comments from Michael Somos, Jul 15 2002

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.

Original entry on oeis.org

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

A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2k,k)x^k/(k + 1)!).

Original entry on oeis.org

1, 1, 3, 12, 59, 343, 2295, 17307, 144751, 1326377, 13189945, 141271298, 1619488645, 19766050827, 255693112641, 3492065507376, 50180426293255, 756444290843433, 11930511611596861, 196404976143077964, 3367697323914503113, 60029614473492823771, 1110430594720934758781

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Exponential transform of A000108.

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 12*x^3/3! + 59*x^4/4! + 343*x^5/5! + 2295*x^6/6! + 17307*x^7/7! + ...

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A001517, A002212, A007317, A080893, A213507, A243953, A302197.

a:=series(exp(add(binomial(2*k,k)*x^k/(k+1)!,k=1..100)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Exp[Sum[CatalanNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] - BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[CatalanNumber[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} A000108(k)*x^k/k!).

E.g.f.: exp(exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)) - 1).

A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).

Original entry on oeis.org

1, 1, 7, 79, 1225, 24121, 575311, 16105447, 517380529, 18752175505, 756760712311, 33645775575391, 1633792107752377, 86022043957561609, 4880923725657950335, 296882100064302393271, 19269430292162925519841, 1329278651404123963041697

Offset: 0

Seiichi Manyama, Aug 02 2024

For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 7, 9, 5, 1, 1, 7, 9, 5, ...], a purely periodic sequence of period 5. Cf. A047974. - Peter Bala, Feb 11 2025

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A375175.

Cf. A004211.

Table[4^n * Sum[Abs[StirlingS1[n, k]] * BellB[k, 1/2] / 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 02 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-4*x)^(1/2)-1)/2)))

a(n) = Sum_{k=0..n} 4^(n-k) * |Stirling1(n,k)| * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 6*(2*n - 3)*a(n-1) - (48*n^2 - 192*n + 191)*a(n-2) + 32*(n-3)*(n-2)*(2*n - 5)*a(n-3).
a(n) ~ 2^(2*n - 1/6) * n^(n - 1/3) / (sqrt(3) * exp(n - 3*2^(-4/3)*n^(1/3) + 1/2)) * (1 - 31/(72*2^(2/3)*n^(1/3)) - 4607/(20736*2^(1/3)*n^(2/3))). (End)
a(n) = (1/exp(1/2)) * (-4)^n * n! * Sum_{k>=0} binomial(-k/2,n)/(2^k * k!). - Seiichi Manyama, Jan 18 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.

A250917 Expansion of e.g.f. exp( xC(x)^3 ) where C(x) = (1 - sqrt(1-4x))/(2*x) is the g.f. of the Catalan numbers, A000108.

Original entry on oeis.org

1, 1, 7, 73, 1033, 18541, 403831, 10351237, 305355793, 10192132153, 379819484551, 15634219476481, 704566985120857, 34506514429777573, 1825081888365736183, 103685565729559782781, 6297505655719537293601, 407233553972252986277617, 27935786938445348562454663

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

In general, if k>0 and e.g.f. = exp(x*C(x)^k) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, then a(n) ~ k * 2^(2*n + k - 5/2) * n^(n-1) / exp(n - 2^(k-2)). - Vaclav Kotesovec, Aug 22 2017

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 73*x^3/3! + 1033*x^4/4! + 18541*x^5/5! +...
such that log(A(x)) = x*C(x)^3,
log(A(x)) = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + 297*x^6 + 1001*x^7 +...
where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Cf. A080893, A251568.
Cf. A091695, A380512, A380515.
Cf. A000108, A001517.

{a(n)=my(C=1); for(i=1, n, C=1+x*C^2 +x*O(x^n));
n!*polcoef(exp(x*C^3), n)}
for(n=0, 20, print1(a(n), ", "))

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(serreverse(x*(1-x))^3/x^2))) \\ Seiichi Manyama, Mar 15 2025

a(n) = Sum_{k=0..n} n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k)  for n>0 with a(0)=1.
a(n) ~ 3 * 2^(2*n+1/2) * n^(n-1) / exp(n-2). - Vaclav Kotesovec, Aug 22 2017
Conjecture D-finite with recurrence: +2*a(n) +(-11*n+20)*a(n-1) +(n^3+9*n^2-116*n+164)*a(n-2) +(-4*n^4+35*n^3+n^2-317*n+342)*a(n-3) -6*(n-3)*(6*n^3-50*n^2+147*n-176)*a(n-4) +12*(n-5)*(2*n-9)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 25 2020
E.g.f.: exp( (1/x)^2 * Series_Reversion( x*(1-x) )^3 ). - Seiichi Manyama, Mar 15 2025

A374882 Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).

Original entry on oeis.org

1, 1, 7, 109, 2665, 88981, 3768391, 193406977, 11663021329, 808092594505, 63252127883431, 5519514702282901, 531266903931402937, 55912682968563924829, 6387276499619184590695, 787104141893585220839401, 104074098535487279656795681, 14697203663694095986066104337

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A252284, A317996.

Cf. A375174, A375176.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1-(1-9*x)^(1/3))/3)))

a(n) = Sum_{k=0..n} (-9)^(n-k) * Stirling1(n,k) * A317996(k) = (-9)^n * Sum_{k=0..n} (1/3)^k * Stirling1(n,k) * Bell_k(-1/3), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 18*(n-2)*a(n-1) - 9*(3*n-8)*(3*n-7)*a(n-2) + a(n-3).
a(n) ~ Gamma(1/3) * 3^(2*n - 3/2) * n^(n - 5/6) / (sqrt(2*Pi) * exp(n - 1/3)) * (1 - 2*Pi/(3^(3/2)*Gamma(1/3)^2*n^(1/3))). (End)

A380640 Expansion of e.g.f. exp(xG(2x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 193, 6673, 319521, 19575001, 1461908449, 128828471073, 13086232224193, 1505486837413801, 193477959856396161, 27472294970916814129, 4271180551913140331233, 721640087945607030774393, 131656978622706616938932641, 25795404137789777215960879681, 5402020596794976601680149234049

Offset: 0

Seiichi Manyama, Jan 28 2025

nonn

Cf. A001764, A380511.
Cf. A380636, A380639.
Cf. A080893, A380643.

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

a(n) = 2 * n! * Sum_{k=0..n-1} 2^k * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
From Vaclav Kotesovec, Jan 29 2025: (Start)
E.g.f. A(x) satisfies x = log(A(x)) * (1 - 2*log(A(x)))^2.
a(n) ~ 3^(3*n - 3/2) * n^(n-1) / (2^(n + 1/2) * exp(n - 1/6)). (End)
a(n) = 2^(n-1)*U(1-n, 2-3*n, 1/2), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
E.g.f.: exp( Series_Reversion( x*(1-2*x)^2 ) ). - Seiichi Manyama, Mar 16 2025

cp's OEIS Frontend

A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.

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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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A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2*k,k)*x^k/(k + 1)!).

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A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).

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Mathematica

PARI

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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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PARI

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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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A250917 Expansion of e.g.f. exp( x*C(x)^3 ) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, A000108.

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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.

A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2k,k)x^k/(k + 1)!).

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.

A250917 Expansion of e.g.f. exp( xC(x)^3 ) where C(x) = (1 - sqrt(1-4x))/(2*x) is the g.f. of the Catalan numbers, A000108.

A380640 Expansion of e.g.f. exp(xG(2x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.