A251568 -id:A251568 - cp's OEIS frontend

A052895 E.g.f.: (1/2)/(exp(x) - 1) * (1 - (5 - 4*exp(x))^(1/2)).

1, 1, 5, 43, 545, 9211, 195305, 4990483, 149371745, 5128125451, 198696086105, 8578228640323, 408387804764945, 21256203702751291, 1200890923560864905, 73191086773679576563, 4786857909878612350145, 334410103752029126714731

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 871

Cf. A000108, A006531, A251568.

spec := [S,{C=Set(Z,1 <= card),S=Sequence(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(1/2)/(E^x-1)*(1-(5-4*E^x)^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
a[n_] = Sum[k! StirlingS2[n, k] CatalanNumber[k], {k, 0, n}];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Jan 15 2018 *)

E.g.f.: (1/2)/(exp(x) - 1)*(1 - (5 - 4*exp(x))^(1/2)).
a(n) = Sum_{k=0..n} k!*Stirling2(n,k)*Catalan(k). - Vladimir Kruchinin, Sep 15 2010
a(n) ~ sqrt(10)*n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
E.g.f.: 1/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 18 2017
From Peter Bala, Jan 15 2018: (Start)
E.g.f.: C(exp(x) - 1), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for A000108. Cf. A006531.
Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 3, 5, 1, 5, 3, 5, ...), with an apparent period 1, 5, 3, 5 of length 4 = phi(10) beginning at a(1). (End)
O.g.f.: 1 + Sum_{k>=1} A000108(k)*Product_{r=1..k} r*x/(1 - r*x). - Petros Hadjicostas, Jun 12 2020

New name using e.g.f. from Vaclav Kotesovec, Sep 30 2013

A380512 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 91, 1753, 45001, 1447471, 56041987, 2539200721, 131859347473, 7723214721271, 503787793244011, 36223369111466857, 2846582772323685721, 242741539845295265503, 22325483241906758894611, 2202979676409063904473121, 232158319570869255177386017, 26024052774273208806612761191

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A001764, A251569, A380511.
Cf. A000262, A251568, A380516.
Cf. A382033.

a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 2*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(3*n,k)/(n-k-1)! for n > 0.
a(n+1) = n! * LaguerreL(n, 2*n+3, -1).
a(n) = (-1)^(n+1)*U(1-n, 2*(1+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

A251569 E.g.f.: exp(xG(x)) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 3, 25, 385, 8661, 255211, 9280573, 401106945, 20075281705, 1141518933811, 72671265032961, 5119905952974913, 395447744211899965, 33224120086567957275, 3016468531370564888101, 294296638636407727046401, 30704676897459478866984273, 3411268107193733242307499235

Offset: 0

Paul D. Hanna, Dec 05 2014

nonn

It appears that a(n) - 1 is divisible by n*(n - 1) for n >= 2. Cf. A251568. - Peter Bala, Feb 15 2015

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 385*x^4/4! + 8661*x^5/5! +...
such that A(x) = exp(x*G(x)) where G(x) = 1 + x*G(x)^3:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

Cf. A001764, A251568.

Flatten[{1,Table[Sum[n!/k! * Binomial[3*n-2*k-1, n-k] * k/(2*n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 15 2015 *)

{a(n)=local(G=1);for(i=1,n,G=1+x*G^3 +x*O(x^n));n!*polcoeff(exp(x*G),n)}
for(n=0,20,print1(a(n),", "))

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

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

Recurrence: 2*(2*n-1)*(54*n^2 - 171*n + 116)*a(n) = (1458*n^4 - 7533*n^3 + 12474*n^2 - 6624*n - 7)*a(n-1) - (324*n^3 - 1080*n^2 + 759*n + 95)*a(n-2) + 8*(n-2)*(54*n^2 - 63*n - 1)*a(n-3). - Vaclav Kotesovec, Feb 15 2015

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

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

A380491 a(n) = n! * Sum_{k=0..n} binomial(2*n-3,k)/(n-k)!.

Original entry on oeis.org

1, 0, 3, 34, 501, 9276, 207775, 5470158, 165625929, 5671386136, 216730118331, 9144481575450, 422249317829053, 21180324426577044, 1146880568461500951, 66677192513929212166, 4142571510546929867025, 273910161452560881843888, 19204878684852222745880179

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380492, A380493.

PARI
```
a(n) = n!*pollaguerre(n, n-3, -1);
```

a(n) = n! * LaguerreL(n, n-3, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n-2).

A380492 a(n) = n! * Sum_{k=0..n} binomial(2*n-2,k)/(n-k)!.

Original entry on oeis.org

1, 1, 7, 73, 1045, 19081, 424051, 11109337, 335262313, 11453449105, 436944953791, 18412283563081, 849345673881277, 42570185481576793, 2303643608370636715, 133859418832759525081, 8312945340897388101841, 549460711493172343519777, 38513032385247860120975863

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380493.

PARI
```
a(n) = n!*pollaguerre(n, n-2, -1);
```

a(n) = n! * LaguerreL(n, n-2, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n-1).

A380493 a(n) = n! * Sum_{k=0..n} binomial(2*n+3,k)/(n-k)!.

Original entry on oeis.org

1, 6, 57, 748, 12585, 259026, 6315001, 178134552, 5711078673, 205209960670, 8171229107481, 357235056697476, 17014791129640057, 877089297426429738, 48657292133825026905, 2890717184573264397616, 183125115830192864360481, 12323226433255671469949622

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380492.

PARI
```
a(n) = n!*pollaguerre(n, n+3, -1);
```

a(n) = n! * LaguerreL(n, n+3, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+4).

A380636 Expansion of e.g.f. exp(xC(2x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.

Original entry on oeis.org

1, 1, 9, 145, 3409, 105921, 4102681, 190630609, 10340890785, 641787925249, 44866443580201, 3489524955627921, 298914951848510449, 27966383049400396225, 2837759948683874979129, 310425081738609550495441, 36418950255827044479693121, 4561668082989623411575958529

Offset: 0

Seiichi Manyama, Jan 28 2025

nonn

Cf. A000108, A251568.

nmax = 20; CoefficientList[Series[E^((1 - Sqrt[1 - 8*x])^2 / (16*x)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 29 2025 *)

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

E.g.f.: exp( (C(2*x)-1)/2 ), where C(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} 2^k * binomial(2*n,k)/(n-k-1)! for n > 0.
a(n+1) = 2^n * n! * LaguerreL(n, n+2, -1/2).
From Vaclav Kotesovec, Jan 29 2025: (Start)
E.g.f.: exp((1 - sqrt(1 - 8*x))^2 / (16*x)).
a(n) ~ 2^(3*n - 1/2) * n^(n-1) / exp(n - 1/2). (End)
a(n) = (-2)^(n-1)*U(1-n, 2+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

A052895 E.g.f.: (1/2)/(exp(x) - 1) * (1 - (5 - 4*exp(x))^(1/2)).

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A380512 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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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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A251569 E.g.f.: exp(x*G(x)) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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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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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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A380491 a(n) = n! * Sum_{k=0..n} binomial(2*n-3,k)/(n-k)!.

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A380492 a(n) = n! * Sum_{k=0..n} binomial(2*n-2,k)/(n-k)!.

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A380512 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

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

A251569 E.g.f.: exp(xG(x)) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

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

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.

A380636 Expansion of e.g.f. exp(xC(2x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.