A001517 -id:A001517 - cp's OEIS frontend

A065919 Bessel polynomial y_n(4).

1, 5, 61, 1225, 34361, 1238221, 54516085, 2836074641, 170218994545, 11577727703701, 880077524475821, 73938089783672665, 6803184337622361001, 680392371852019772765, 73489179344355757819621, 8525425196317119926848801, 1057226213522667226687070945

Offset: 0

N. J. A. Sloane, Dec 08 2001

Main diagonal of A143411. - Peter Bala, Aug 14 2008

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Harry J. Smith, Table of n, a(n) for n = 0..100
W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001517, A001518.
Cf. A143411 (main diagonal), A143412.
Polynomial coefficients are in A001498.

A065919:= func< n | (&+[Binomial(n,k)*Factorial(n+k)*2^k/Factorial(n): k in [0..n]]) >;
[A065919(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

seq(simplify(2^n*KummerU(-n,-2*n,1/2)), n=0..16); # Peter Luschny, May 10 2022

Table[Sum[(n+k)!*2^k/((n-k)!*k!), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)

for (n=0, 100, if (n>1, a=4*(2*n - 1)*a1 + a2; a2=a1; a1=a, if (n, a=a1=5, a=a2=1)); write("b065919.txt", n, " ", a) ) \\ Harry J. Smith, Nov 04 2009

a(n) = sum(k=0,n, (n+k)!*2^k/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013

def A065919(n): return sum(binomial(n,k)*factorial(n+k)*2^k/factorial(n) for k in range(n+1))
[A065919(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
From Peter Bala, Aug 14 2008: (Start)
Recurrence relation: a(0) = 1, a(1) = 5, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A143412(n) satisfies the same recurrence relation.
1/sqrt(e) = 1 - 2*Sum_{n = 0..inf} (-1)^n/(a(n)*a(n+1)) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). (End)
G.f.: 1/Q(0), where Q(k)= 1 - x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = exp(1/4)/sqrt(2*Pi)*BesselK(n+1/2,1/4). - Gerry Martens, Jul 22 2015
a(n) ~ 2^(3*n+1/2) * n^n / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015
From Peter Bala, Apr 12 2017: (Start)
a(n) = 1/n!*Integral_{x = 0..inf} x^n*(1 + 2*x)^n dx.
E.g.f.: d/dx( exp(x*c(2*x)) ) = 1 + 5*x + 61*x^2/2! + 1225*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
G.f.: (1/(1-x))*hypergeometric2f0(1,1/2; - ; 8*x/(1-x)^2). - G. C. Greubel, Aug 16 2017
a(n) = 2^n*KummerU(-n, -2*n, 1/2). - Peter Luschny, May 10 2022

A099022 a(n) = Sum_{k=0..n} C(n,k)(2n-k)!.

Original entry on oeis.org

1, 3, 38, 1158, 65304, 5900520, 780827760, 142358474160, 34209760152960, 10478436416945280, 3984884716852972800, 1842169367191937414400, 1017403495472574045158400, 661599650478455071589606400, 500354503197888042597961267200, 435447353708763072625260119808000

Offset: 0

Ralf Stephan, Sep 23 2004

Diagonal of Euler-Seidel matrix with start sequence n!.
Number of ways to use the elements of {1,..,k}, n<=k<=2n, once each to form a sequence of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006
Replace "lists" by "sets": A105749.

Robert Israel, Table of n, a(n) for n = 0..224
L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (9).
Index entries for related partition-counting sequences

Cf. A001517, A076571, A082765 (binomial transform), A105749, row sums of A328826.

f:= gfun:-rectoproc({a(n)=2*n*(2*n-1)*a(n-1)+n*(n-1)*a(n-2), a(0)=1,a(1)=3},a(n),remember):
map(f, [$0..20]); # Robert Israel, Feb 15 2017

Table[(2k)! Hypergeometric1F1[-k, -2k, 1], {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
Table[Sum[Binomial[n,k](2n-k)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Nov 22 2021 *)

for(n=0,25, print1(sum(k=0,n, binomial(n,k)*(2*n-k)!), ", ")) \\ G. C. Greubel, Dec 31 2017

T(2*n, n), where T is the triangle in A076571.
a(n) = n!*A001517(n).
A082765(n) = Sum[C(n, k)*a(k), 0<=k<=n].
a(n) = 2*n*(2*n-1)*a(n-1)+n*(n-1)*a(n-2). - Vladeta Jovovic, Sep 27 2004
a(n) = int {x = 0..inf} exp(-x)*(x + x^2)^n dx. Applying the results of Nicolaescu, Section 3.2 to this integral we obtain the asymptotic expansion a(n) ~ (2*n)!*exp(1/2)*( 1 - 1/(16*n) - 191/(6144*n^2) + O(1/n^3) ). - Peter Bala, Jul 07 2014

A113025 Triangle of integer coefficients of polynomials P(n,x) of degree n, and falling powers of x, arising in diagonal Padé approximation of exp(x).

Original entry on oeis.org

1, 1, 2, 1, 6, 12, 1, 12, 60, 120, 1, 20, 180, 840, 1680, 1, 30, 420, 3360, 15120, 30240, 1, 42, 840, 10080, 75600, 332640, 665280, 1, 56, 1512, 25200, 277200, 1995840, 8648640, 17297280, 1, 72, 2520, 55440, 831600, 8648640, 60540480, 259459200

Offset: 0

Benoit Cloitre, Jan 03 2006

exp(x) is well approximated by P(n,x)/P(n,-x). (P(n,1)/P(n,-1))_{n>=0} is a sequence of convergents to e: i.e., P(n,1) = A001517(n) and P(n,-1) = abs(A002119(n)).
From Roger L. Bagula, Feb 15 2009: (Start)
The row polynomials in rising powers of x are y_n(2*x) = Sum_{k=0..n} binomial(n+k, 2*k)*((2*k)!/k!)*x^k, for n >= 0, with the Bessel polynomials y_n(x) of Krall and Frink, eq. (3), (see also Grosswald, p. 18, eq. (7) and Riordan, p. 77). For the coefficients see A001498. [Edited by Wolfdieter Lang, May 11 2018]
P(n, x) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!)*x^(n-k).
Row sums are A001517. (End)

			P(3,x) = x^3 + 12*x^2 + 60*x + 120.
y_3(2*x) = 1 + 12*x + 60*x^2 + 120*x^3. (Bessel with x -> 2*x).
From _Roger L. Bagula_, Feb 15 2009: (Start)
{1},
{1, 2},
{1, 6, 12},
{1, 12, 60, 120},
{1, 20, 180, 840, 1680},
{1, 30, 420, 3360, 15120, 30240},
{1, 42, 840, 10080, 75600, 332640, 665280},
{1, 56, 1512, 25200, 277200, 1995840, 8648640, 17297280},
{1, 72, 2520, 55440, 831600, 8648640, 60540480, 259459200, 518918400},
{1, 90, 3960, 110880, 2162160, 30270240, 302702400, 2075673600, 8821612800, 17643225600},
{1, 110, 5940, 205920, 5045040, 90810720, 1210809600, 11762150400, 79394515200, 335221286400, 670442572800} (End)

J. Riordan, Combinatorial Identities, Wiley, 1968, p.77, 10. [From Roger L. Bagula, Feb 15 2009]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
E. Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. vol. 698, 1978, p. 18.
H. L. Krall and Orrin Frink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
Eric Weisstein's World of Mathematics, Padé approximants.
F. Wielonsky, Asymptotics of diagonal Hermite-Pade approximants to exp(x), J. Approx. Theory 90 (1997) 283-298.

Cf. A001498, A001517, A303986 (signed version).

T := (n, k) -> pochhammer(n+1, k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, May 11 2018

L[n_, m_] = (n + m)!/((n - m)!*m!);
Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula, Feb 15 2009 *)
P[x_, n_] := Sum[ (2*n - k)!/(k!*(n - k)!)*x^(k), {k, 0, n}]; Table[Reverse[CoefficientList[P[x, n], x]], {n,0,10}] // Flatten (* G. C. Greubel, Aug 15 2017 *)

PARI
```
T(n,k)=(n+k)!/k!/(n-k)!
```

From Wolfdieter Lang, May 11 2018: (Start)
T(n, k) = binomial(n+k, 2*k)*(2*k)!/k! = (n+k)!/((n-k)!*k!), n >= 0, k = 0..n. (see the R. L. Baluga comment above).
Recurrence (adapted from A001498, see the Grosswald reference): For n >= 0, k = 0..n: a(n, k) = 0 for n < k (zeros not shown in the triangle), a(n, -1) = 0, a(0, 0) = 1 = a(1, 0) and otherwise a(n, k) = 2*(2*n-1)*a(n-1, k-1) + a(n-2, k).
(End)
T(n, k) = Pochhammer(n+1, k)*binomial(n, k). # Peter Luschny, May 11 2018

A053987 Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).

Original entry on oeis.org

1, 6, 59, 820, 14701, 322602, 8372951, 250865928, 8521068601, 323549740910, 13580568049619, 624382580541564, 31205548459028581, 1684475234207001810, 97668358035547076399, 6053753722969711734928, 399450077357965427428849, 27955451661334610208284502

Offset: 1

Vladeta Jovovic, Apr 03 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..200
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Cf. A001517, A053988, A058798.

A053987:= func< n| &+[(-1)^k*Factorial(2*n-2*k-1)/(Factorial(n-2*k-1)* Factorial(2*k+1)): k in [0..Floor((n-1)/2)]] >;
[A053987(n) : n in [1..20]]; // G. C. Greubel, May 17 2020

A053987 := n -> local k; add((-1)^k*(2*n-2*k-1)!/((n-2*k-1)!*(2*k+1)!), k = 0..floor((n-1)/2)); seq(A053987(n), n = 1..20); # G. C. Greubel, May 17 2020

Rest[CoefficientList[Series[Sin[(1-Sqrt[1-4*x])/2]/Sqrt[1-4*x], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 25 2014 *)

a(n)=sum(k=0,floor((n-1)/2),(-1)^k*(2*n-2*k-1)!/(n-2*k-1)!/(2*k+1)!) \\ Benoit Cloitre, Jan 03 2006

def A053987(n):
    return 4^n*gamma(n+1/2)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], -1/4)/sqrt(4*pi)
[round(A053987(n).n(100)) for n in (1..18)] # Peter Luschny, Sep 10 2014

a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k*(2*n-2*k-1)!/((n-2*k-1)! * (2*k+1)!). - Benoit Cloitre, Jan 03 2006
E.g.f.: 1-cos(x*C(x)), C(x)=(1-sqrt(1-4*x))/(2*x) (A000108). - Vladimir Kruchinin, Aug 10 2010
From Peter Bala, Aug 01 2013, (Start)
a(n+1) = (4*n+2)*a(n) - a(n-1) with a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*4^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1, k)*binomial(n-k-1/2, k+1/2), see A058798. (End)
a(n) ~ sin(1/2) * 2^(2*n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Feb 25 2014
a(n) = 4^n*Gamma(n+1/2)*hypergeometric([1/2-n/2,1-n/2], [3/2,1/2-n,1-n], -1/4)/sqrt(4*Pi). - Peter Luschny, Sep 10 2014

a(16)-a(17) from Wesley Ivan Hurt, Feb 28 2014

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).

A079165 a(n) = (4n-2)*a(n-1)+a(n-2) with a(0)=1 and a(1)=2.

Original entry on oeis.org

1, 2, 13, 132, 1861, 33630, 741721, 19318376, 580293001, 19749280410, 751052948581, 31563973120812, 1452693816505933, 72666254798417462, 3925430452931048881, 227747632524799252560, 14124278646990484707601

Offset: 0

Henry Bottomley, Dec 31 2002

nonn

			a(3) = (4*3-2)*a(2)+a(1) = 10*13+2 = 132.

Robert Israel, Table of n, a(n) for n = 0..365

Cf. A001517, A002119, A079166.

f:= gfun:-rectoproc({a(n)=(4*n-2)*a(n-1)+a(n-2),a(0)=1,a(1)=2},a(n),remember):
map(f, [$0..50]); # Robert Israel, May 03 2016

a[n_] := Sum[(2n-2k)!/((n-2k)! (2k)!), {k, 0, n/2}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 10 2019, after Vladimir Kruchinin *)

a(n):=sum((2*n-2*k)!/((n-2*k)!*(2*k)!),k,0,n/2); /* Vladimir Kruchinin, May 03 2016 */

a(n) = (A001517(n)+|A002119(n)|)/2 = A079166(2, n). a(n)/|A002119(n)| tends to 1.8591409...=(e+1)/2; a(n)/A001517(n) tends to 0.68393972...=2e/(e+1).
E.g.f.: cosh((1-sqrt(1-4*x))/2)/sqrt(1-4*x). - Vladimir Kruchinin, May 03 2016
a(n) = Sum_{k=0..n/2}((2*n-2*k)!/((n-2*k)!*(2*k)!)). - Vladimir Kruchinin, May 03 2016
a(n) = ((-1)^n*sqrt(Pi*exp(-1))*BesselI((2*n+1)/2, 1/2))/2 + (BesselK((2*n+1)/2, 1/2)*cosh(1/2))/sqrt(Pi), where BesselI(n,x) is the modified Bessel function of the first kind, BesselK(n,x) is the modified Bessel function of the second kind. - Ilya Gutkovskiy, May 03 2016
a(n) = (hypergeom([-n,n+1],[],-1)+(-1)^n*hypergeom([-n,n+1],[],1))/2. - Peter Luschny, May 03 2016

A105747 Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) lists, each of length at most 2.

Original entry on oeis.org

1, 4, 23, 216, 2937, 52108, 1136591, 29382320, 877838673, 29753600404, 1127881002535, 47278107653768, 2171286661012617, 108417864555606300, 5847857079417024031, 338841578119273846112

Offset: 0

Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005

			a(2)=23:
{(),()},
{(),(1)},
{(),(1,2)},
{(),(2,1)},
{(1),(2)},
{(1),(2,3)},
{(1),(3,2)},
...,
{(1,4),(2,3)},
{(1,4),(3,2)},
{(4,1),(2,3)},
{(4,1),(3,2)}.

Seiichi Manyama, Table of n, a(n) for n = 0..365
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions! arXiv math.CO.0606404.
Index entries for related partition-counting sequences

First differences: A001517.
Replace "collection" by "sequence": A082765.
Replace "lists" by "sets": A105748.

Table[Sum[(k+i)!/i!/(k-i)!, {k, 0, n}, {i, 0, k}], {n, 0, 20}]

a(n) = Sum_{0<=i<=k<=n} (k+i)!/i!/(k-i)!.
a(n+3) = (4*n+11)*a(n+2) - (4*n+9)*a(n+1) - a(n) - Benoit Cloitre, May 26 2006
G.f.: 1/(1-x)/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) ~ 2^(2*n + 1/2) * n^n / exp(n - 1/2). - Vaclav Kotesovec, May 15 2022

A119274 Triangle of coefficients of numerators in Padé approximation to exp(x).

Original entry on oeis.org

1, 2, 1, 12, 6, 1, 120, 60, 12, 1, 1680, 840, 180, 20, 1, 30240, 15120, 3360, 420, 30, 1, 665280, 332640, 75600, 10080, 840, 42, 1, 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1, 518918400, 259459200, 60540480, 8648640, 831600, 55440, 2520

Offset: 0

Paul Barry, May 12 2006

n-th numerator of Padé approximation is (1/n!)*sum{j=0..n, C(n,j)(2n-j)!x^j}. Reversal of A113025. Row sums are A001517. First column is A001813. Inverse is A119275.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
Dividing each diagonal by its initial element generates A054142. - Tom Copeland, Oct 10 2016

			Triangle begins
1,
2, 1,
12, 6, 1,
120, 60, 12, 1,
1680, 840, 180, 20, 1,
30240, 15120, 3360, 420, 30, 1

Cf. A001497, A054142.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (2*n)!/n!, 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(2#)!/#!&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[bell_transform from A264428]
# Adds a column 1,0,0,0,... at the left side of the triangle.
def A119274_row(n):
    multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
    mfact = [multifact_4_2(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A119274_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

Number triangle T(n,k) = C(n,k)(2n-k)!/n!.

After adding a leading column (1,0,0,0,...), the triangle gives the coefficients of the Sheffer associated sequence (binomial-type polynomials) for the delta (lowering) operator D(1-D) with e.g.f. exp[ x * (1 - sqrt(1-4t)) / 2 ] . See Mathworld on Sheffer sequences. See A134685 for relation to Catalan numbers. - Tom Copeland, Feb 09 2008

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

A369722 Expansion of e.g.f. exp( (3/2) * (1-sqrt(1-4*x)) ).

Original entry on oeis.org

1, 3, 15, 117, 1305, 19323, 359559, 8084205, 213425361, 6475518675, 222088463199, 8497641269637, 358899729493545, 16585866328129803, 832523413971932055, 45105537151437499197, 2623613865509122341921, 163070009495928522691875

Offset: 0

Seiichi Manyama, Jan 30 2024

nonn

Cf. A001517, A101682, A243953, A369723, A369724.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3/2*(1-sqrt(1-4*x)))))

a(0) = 1; a(n) = Sum_{k=0..n-1} 3^(n-k) * (n-1+k)! / (k! * (n-1-k)!).

a(n) = 2*(2*n-3)*a(n-1) + 9*a(n-2).

cp's OEIS Frontend

A065919 Bessel polynomial y_n(4).

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A099022 a(n) = Sum_{k=0..n} C(n,k)*(2*n-k)!.

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A113025 Triangle of integer coefficients of polynomials P(n,x) of degree n, and falling powers of x, arising in diagonal Padé approximation of exp(x).

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A053987 Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).

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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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A079165 a(n) = (4n-2)*a(n-1)+a(n-2) with a(0)=1 and a(1)=2.

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A099022 a(n) = Sum_{k=0..n} C(n,k)(2n-k)!.

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

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.