A047974 -id:A047974 - cp's OEIS frontend

A115329 Expansion of e.g.f.: exp(x + 2*x^2).

1, 1, 5, 13, 73, 281, 1741, 8485, 57233, 328753, 2389141, 15539261, 120661465, 866545993, 7140942173, 55667517781, 484124048161, 4046845186145, 36967280461093, 328340133863533, 3137853448906601, 29405064157989241

Offset: 0

Paul D. Hanna, Jan 20 2006

Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.
Combinatorial interpretation: a(n) counts the partitions of a set of n distinguishable objects into subsets of size 1 and 2 with the additional feature that the constituents of the subset of size 2 acquire 2 colors. - Karol A. Penson and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006
In general, e.g.f. exp(x+m*x^2) has general term sum{k=0..n, C(n,k)*m^k*(n-k)!/(n-m*k)!}. [Paul Barry, Nov 07 2008]
The sequence terms have the form 4*m + 1 (follows from the recurrence). a(n+k) = a(n) (mod k) holds for all n and k by an induction argument making use of the recurrence equation. For each k the sequence a(n) taken modulo k is thus periodic with exact period dividing k. - Peter Bala, Nov 15 2017

Seiichi Manyama, Table of n, a(n) for n = 0..665 (terms 0..200 from Vincenzo Librandi)
Magdalena Boos, Giovanni Cerulli Irelli, Francesco Esposito, Parabolic orbits of 2-nilpotent elements for classical groups, arXiv:1802.06425 [math.RT], 2018.

Column k=4 of A359762.
Cf. A000085, A047974, A115330.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), this sequence (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x+2*x^2) ))); // G. C. Greubel, Jul 12 2024

a := n -> I^(1 - n)*2^((3*(n - 1))/2)*KummerU((1 - n)/2, 3/2, -1/8):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Nov 21 2017

Range[0, 20]! CoefficientList[Series[Exp[(x + 2 x^2)], {x, 0, 20}], x] (* Vincenzo Librandi, May 22 2013 *)

a(n)=local(m=4);n!*polcoeff(exp(x+m/2*x^2+x*O(x^n)),n)

[(-i*sqrt(2))^n*hermite(n, i/(2*sqrt(2))) for n in range(31)] # G. C. Greubel, Jul 12 2024

Term-by-term square equals A115330 which has e.g.f.: exp(x/(1-4*x))/sqrt(1-16*x^2).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)2^k*n!/(n-k)! = Sum_{k=0..n} C(n,k)2^k*(n-k)!/(n-2k)!. - Paul Barry, Nov 07 2008
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A000085 and A047974. - Peter Bala, Dec 07 2011
a(n) = a(n-1) + 4*(n-1)*a(n-2). - R. J. Mathar, Dec 10 2011
a(n) ~ 2^(n-1/2)*exp(sqrt(n)/2-n/2-1/16)*n^(n/2). - Vaclav Kotesovec, Oct 19 2012
G.f.: 1/Q(0), where Q(k)= 1 + 4*x*k - x/(1 - 4*x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013
G.f.: 1/G(0), where G(k)= 1 - x - 4*(k+1)*x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 21 2013
a(n) = i^(1 - n)*2^(3*(n - 1)/2)*KummerU((1 - n)/2, 3/2, -1/8). - Peter Luschny, Nov 21 2017
a(n) = (-i*sqrt(2))^n * Hermite(n, i/(2*sqrt(2))). - G. C. Greubel, Jul 12 2024

More terms from Karol A. Penson and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006

A080893 Expansion of e.g.f. exp(xC(x)) = exp((1-sqrt(1-4x))/2), where C(x) is the g.f. of the Catalan numbers A000108.

Original entry on oeis.org

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

Offset: 0

Emanuele Munarini, Mar 31 2003

Essentially the same as A001517: a(n+1) = A001517(n).

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, 3, 9, 3, 1, 1, 3, 9, 3, ...], a purely periodic sequence of period 5. Cf. A047974. - Peter Bala, Feb 11 2025

W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000108, A001517, A009235, A375173.

y[x_] := y[x] = 2(2x - 3)y[x - 1] + y[x - 2]; y[0] = 1; y[1] = 1; Table[y[n],{n,0,17}]
With[{nn=20},CoefficientList[Series[Exp[(1-Sqrt[1-4x])/2],{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Oct 30 2011 *)

{a(n) = if( n<1, n = 1 - n); n! * polcoeff( exp( (1 - sqrt(1 - 4*x + x * O(x^n))) / 2), n)} /* Michael Somos, Apr 07 2012 */

A080893 = lambda n: hypergeometric([-n+1, n], [], -1)
[simplify(A080893(n)) for n in (0..19)] # Peter Luschny, Oct 17 2014

E.g.f.: exp((1-sqrt(1-4*x))/2).
D-finite with recurrence: a(n+2) = 2*(2*n + 1)*a(n+1) + a(n).
Recurrence: y(n+1) = Sum_{k = 0..n} binomial(n, k)*binomial(2k, k)*k!*y(n-k).
a(1 - n) = a(n).  a(n + 1) = A001517(n). - Michael Somos, Apr 07 2012
G.f.: 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-3/2)*n^(n-1)/exp(n-1/2). - Vaclav Kotesovec, Jun 26 2013
a(n) = hypergeom([-n+1, n], [], -1). - Peter Luschny, Oct 17 2014
a(n) = Sum_{k=0..n} (-4)^(n-k) * Stirling1(n,k) * A009235(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. - Seiichi Manyama, Aug 02 2024

A143405 Number of forests of labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, also row sums of A143395, A143396 and A143397.

Original entry on oeis.org

1, 1, 4, 17, 89, 552, 3895, 30641, 265186, 2497551, 25373097, 276105106, 3199697517, 39297401197, 509370849148, 6943232742493, 99217486649933, 1482237515573624, 23093484367004715, 374416757914118941, 6304680593346141746, 110063311977033807187

Offset: 0

Alois P. Heinz, Aug 12 2008

nonn

a(n) is the number of the partitions of an n-set where each block is endowed with a nonempty subset. - Emanuele Munarini, Sep 15 2016

			a(2) = 4, because there are 4 forests for 2 labels: {1,2}, {1}{2}, {1}<-2, {2}<-1.

Alois P. Heinz, Table of n, a(n) for n = 0..504
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Vaclav Kotesovec, Asymptotics of OEIS A143405, A355291 and the generalization, Jul 12 2022 (this version also includes several figures).
Index entries for sequences related to rooted trees

Cf. A055882, A143395, A143396, A143397, A048993, A008277, A007318, A000110, A355291.

a:= n-> add(add(binomial(n, t)*Stirling2(t, k)*k^(n-t), t=k..n), k=0..n):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 05 2019

CoefficientList[Series[Exp[Exp[t] (Exp[t] - 1)], {t, 0, 12}], t] Range[0, 12]! (* Emanuele Munarini, Sep 15 2016 *)
Table[Sum[Binomial[n, k] 2^k BellB[k] BellB[n - k, -1], {k, 0, n}], {n, 0, 12}] (* Emanuele Munarini, Sep 15 2016 *)
Table[Sum[BellY[n, k, 2^Range[n] - 1], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n) = sum(k=0, n, k!*sum(j=0, k\2, 1/(j!*(k-2*j)!))*stirling(n, k, 2)); \\ Seiichi Manyama, May 14 2022

a(n) = Sum_{k=0..n} Sum_{t=k..n} C(n,t) * Stirling2(t,k)*k^(n-t).
a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k) * Stirling2(k,t)*t^(n-k).
a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k-t) * Stirling2(n-(k-t),t)*t^(k-t).
E.g.f.: exp(exp(x)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008
a(n) = sum(binomial(n,k)*2^k*bell(k)*S(n-k,-1),k=0..n), where bell(n) are the Bell numbers (A000110) and S(n,x) = sum(Stirling2(n,k)*x^k,k=0..n) are the Stirling (or exponential) polynomials. - Emanuele Munarini, Sep 15 2016
Identity: sum(binomial(n,k)*a(k)*bell(n-k),k=0..n) = 2^n*bell(n). - Emanuele Munarini, Sep 15 2016
a(n) = Sum_{k=0..n} A047974(k) * Stirling2(n,k). - Seiichi Manyama, May 14 2022
a(n) ~ exp(exp(2*z) - exp(z) - n) * (n/z)^(n + 1/2) / sqrt(2*(1 + 2*z)*exp(2*z) - (1 + z)*exp(z)), where z = LambertW(n)/2 - 1/(1 + 2/LambertW(n) - 4 * n^(1/2) * (1 + LambertW(n)) / LambertW(n)^(3/2)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ 2^n * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n * exp(n + 1/8 - n/LambertW(n) + sqrt(n/LambertW(n)))). - Vaclav Kotesovec, Jul 08 2022

A158954 Numerator of Hermite(n, 1/4).

Original entry on oeis.org

1, 1, -7, -23, 145, 881, -4919, -47207, 228257, 3249505, -13184999, -273145399, 887134513, 27109092817, -65152896535, -3101371292039, 4716976292161, 401692501673153, -239816274060743, -58083536514994775, -21631462857761839, 9271734379541402161

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 1/2, -7/4, -23/8, 145/16, 881/32, -4919/64, -47207/128, 228257/256, 3249505/512, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Cf. A000079 (denominators).

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), this sequence (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

[Numerator((&+[(-1)^k*Factorial(n)*(1/2)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018

A158954 := proc(n)
        orthopoly[H](n,1/4) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator[Table[HermiteH[n,1/4],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 23 2011 *)

a(n)=numerator(polhermite(n,1/4)) \\ Charles R Greathouse IV, Jan 29 2016

[2^n*hermite(n, 1/4) for n in range(31)] # G. C. Greubel, Jul 12 2024

D-finite with recurrence a(n) - a(n-1) + 8*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 16 2014
From G. C. Greubel, Jun 09 2018: (Start)
a(n) = 2^n*Hermite(n,1/4).
E.g.f.: exp(x-4*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/2)^(n-2k)/(k!*(n-2k)!)). (End)

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 1, 5, 25, 193, 1481, 16021, 167665, 2220065, 30004273, 468585541, 7560838121, 138355144225, 2589359765305, 53501800316693, 1146089983207681, 26457132132638401, 632544682981967585, 16171678558995779845, 426926324177655018553, 11938570457328874969601

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 13 2017: (Start)
The terms of the sequence appear to be of the form 4*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..438
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).

Cf. A000005, A028342, A295739, A006171, A107742.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 05 2018 *)
a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 06 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A118395 Expansion of e.g.f. exp(x + x^3).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 481, 2731, 10417, 91225, 681121, 3493711, 33597961, 303321877, 1938378625, 20282865331, 211375647841, 1607008257841, 18157826367937, 212200671085975, 1860991143630841, 22560913203079021, 289933758771407521, 2869267483843753147, 37116733726117707025

Offset: 0

Paul D. Hanna, May 07 2006

nonn

Equals row sums of triangle A118394.

Robert Israel, Table of n, a(n) for n = 0..533

Cf. A118394, A118396.

Cf. A047974, A190875, A190877.

[n le 3 select 1 else Self(n-1) + 3*(n-2)*(n-3)*Self(n-3): n in [1..26]]; // Vincenzo Librandi, Aug 25 2015

with(combstruct):seq(count(([S, {S=Set(Union(Z, Prod(Z, Z, Z)))}, labeled], size=n)), n=0..22); # Zerinvary Lajos, Mar 18 2008

CoefficientList[Series[E^(x+x^3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)
T[n_, k_] := n!/(k!(n-3k)!);
a[n_] := Sum[T[n, k], {k, 0, Floor[n/3]}];
a /@ Range[0, 24] (* Jean-François Alcover, Nov 04 2020 *)

a(n)=n!*polcoeff(exp(x+x^3+x*O(x^n)),n)

N=33;  x='x+O('x^N);
egf=exp(x+x^3);
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */

a(n) = n!*sum(k=0, n\3, binomial(n-2*k, k)/(n-2*k)!); \\ Seiichi Manyama, Feb 25 2022

def a(n):
    if (n<3): return 1
    else: return a(n-1) + 3*(n-1)*(n-2)*a(n-3)
[a(n) for n in (0..25)] # G. C. Greubel, Feb 18 2021

E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) =  1 + (1+x^2)/(k+1)/(1-x/(x+(1)/G(k+1) )), recursively defined continued fraction. - Sergei N. Gladkovskii, Feb 04 2013
a(n) ~ 3^(n/3-1/2) * n^(2*n/3) * exp((n/3)^(1/3)-2*n/3). - Vaclav Kotesovec, Jun 02 2013
E.g.f.: A(x) = exp(x+x^3) satisfies A' - (1+3*x^2)*A = 0. - Gheorghe Coserea, Aug 24 2015
a(n+1) = a(n) + 3*n*(n-1)*a(n-2). - Gheorghe Coserea, Aug 24 2015
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,k)/(n-2*k)!. - Seiichi Manyama, Feb 25 2022

Missing a(0)=1 prepended by Joerg Arndt, Sep 15 2012

A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

Original entry on oeis.org

1, 1, 3, 19, 121, 1041, 10651, 121843, 1575729, 22970881, 366805171, 6365365491, 120044573353, 2430782532049, 52677233993931, 1217023986185491, 29799465317716321, 771272544315151233, 21044341084622337379, 603173026772647474771

Offset: 0

Seiichi Manyama, Oct 17 2017

From Peter Bala, Mar 28 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 3, 9, 1, 1, 1, 3, 9, 1, ...], a purely periodic sequence with exact period 5.
3 divides a(3*n+2); 9 divides a(9*n+8); 11 divides a(11*n+4); 19 divides a(19*n+3). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A293839, A293841.

Cf. A058892.

nmax = 20; CoefficientList[Series[E^Sum[PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k)*a(n-k)/(n-k)! for n > 0.

A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

Original entry on oeis.org

1, -1, -1, 5, 1, -41, 31, 461, -895, -6481, 22591, 107029, -604031, -1964665, 17669471, 37341149, -567425279, -627491489, 19919950975, 2669742629, -759627879679, 652838174519, 31251532771999, -59976412450835, -1377594095061119, 4256461892701199, 64623242860354751

Offset: 0

N. J. A. Sloane

Binomial transform gives A067994. Inverse binomial transform gives A062267(n)*(-1)^n. - Vladimir Reshetnikov, Oct 11 2016

The congruence a(n+k) == (-1)^k*a(n) (mod k) holds for all n and k. It follows that for even k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of k, while for odd k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of 2*k. See A047974. - Peter Bala, Apr 10 2023

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 209.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..732 (terms 0..200 from T. D. Noe)
Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.
Index entries for sequences related to Hermite polynomials

Cf. A000186, A062267, A144141, A293604.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

Table[HermiteH[n, -1/2], {n, 0, 25}] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[(-2)^n HypergeometricU[-n/2, 1/2, 1/4], {n, 0, 25}] (* Benedict W. J. Irwin, Oct 17 2017 *)

N=66;  x='x+O('x^N);
egf=exp(-x-x^2);  Vec(serlaplace(egf))
/* Joerg Arndt, Mar 07 2013 */

vector(50, n, n--; sum(k=0, n/2, (-1)^(n-k)*k!*binomial(n, k)*binomial(n-k, k))) \\ Altug Alkan, Oct 22 2015

a(n) = polhermite(n, -1/2); \\ Michel Marcus, Oct 12 2016

from sympy import hermite
def a(n): return hermite(n, -1/2) # Indranil Ghosh, May 26 2017

E.g.f.: exp(-x-x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*k!*C(n, k)*C(n-k, k).
a(n) = - a(n-1) - 2*(n-1)*a(n-2), a(0) = 1, a(1) = -1.
A000186(n) ~ n!^2*exp(1)^(-3)*(a(0) + a(1)/n + a(2)/(2*[n]2) + ... + a(k)/(k!*[n]_k) + ...), where [n]_k = n*(n-1)*...*(n-k + 1), [n]_0 = 1. - _Vladeta Jovovic, Apr 30 2001
a(n) = Sum_{k=0..n} (-1)^(2*n-k)*C(k,n-k)*n!/k!. - Paul Barry, Oct 08 2007, corrected by Altug Alkan, Oct 22 2015
E.g.f.: 1 - x*(1 - E(0) )/(1+x) where E(k) = 1 - (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
E.g.f.: -x/Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 1/(x*Q(0)), where Q(k) = 1 + 1/x + 2*(k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013
a(n) = (-2)^n * U(-n/2, 1/2, 1/4), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017
E.g.f.: Product_{k>=1} (1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019

Formulae and more terms from Vladeta Jovovic, Apr 30 2001

A018191 a(n) = Sum_{k=0..n} binomial(n, k) * k! / floor(k/2)!.

Original entry on oeis.org

1, 2, 5, 16, 53, 206, 817, 3620, 16361, 80218, 401501, 2139512, 11641885, 66599846, 388962953, 2367284236, 14700573137, 94523836850, 619674301621, 4186249123808, 28809504493061, 203556335785342, 1463877667140065, 10777146970619636, 80686484464418233

Offset: 0

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

nonn

Binomial transform of { n!/floor(n/2)! }.
Number of symmetric chord diagrams of degree n-1.
Row sums of exponential Riordan array [(1+x), x(1+x)]. - Paul Barry, Apr 17 2007

N. J. A. Sloane, Table of n, a(n) for n = 0..300 (first 200 terms from _Vincenzo Librandi_)
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233.

Cf. A047974, A081125.

f:=n-> add(binomial(n,k)*k!/floor(k/2)!, k=0..n); [seq(f(n),n=1..40)]; # N. J. A. Sloane, Sep 25 2021

a[n_] := Sum[Binomial[n-1, k] k! / Floor[k/2]!, {k, 0, n}];
Array[a, 25] (* Jean-François Alcover, Aug 29 2019 *)
Table[n!*SeriesCoefficient[(1+x)*E^(x+x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

a(n) = A047974(n-1) + (n-1)*A047974(n-2). - Vladeta Jovovic, Aug 06 2006
E.g.f.: (1 + x)*exp(x + x^2). - Vladeta Jovovic, Aug 06 2006
Recurrence: (n-2)*a(n) = (n-3)*a(n-1) + 2*(n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ 2^(n/2 - 1)*exp(sqrt(n/2) - n/2 - 1/8)*n^(n/2 + 1/2)*(1 + 85/96*sqrt(2)/sqrt(n)). - Vaclav Kotesovec, Oct 13 2012
a(n) = -(n-3)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2. - Seiichi Manyama, Nov 12 2024

Entry revised by N. J. A. Sloane, Sep 25 2021

A293720 Expansion of e.g.f.: exp(x + 4*x^2).

Original entry on oeis.org

1, 1, 9, 25, 241, 1041, 10681, 60649, 658785, 4540321, 51972841, 415198521, 4988808529, 44847866545, 563683953561, 5586645006601, 73228719433921, 788319280278849, 10747425123292105, 124265401483446361, 1757874020223846321, 21640338257575264081

Offset: 0

Seiichi Manyama, Oct 15 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..612

Column k=2 of A293724.
Column k=8 of A359762.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), this sequence (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x+4*x^2) ))); // G. C. Greubel, Jul 12 2024

CoefficientList[Series[E^(x + 4*x^2), {x,0,30}], x] * Range[0,30]! (* Vaclav Kotesovec, Oct 15 2017 *)

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x+4*x^2)))

[(-2*i)^n*hermite(n, i/4) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) ~ 2^((3*n-1)/2) * exp(-1/32 + sqrt(2*n)/4 - n/2) * n^(n/2). - Vaclav Kotesovec, Oct 15 2017

a(n) = (-2*i)^n * Hermite(n, i/4). - G. C. Greubel, Jul 12 2024

cp's OEIS Frontend

A115329 Expansion of e.g.f.: exp(x + 2*x^2).

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A080893 Expansion of e.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.

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A143405 Number of forests of labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, also row sums of A143395, A143396 and A143397.

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A158954 Numerator of Hermite(n, 1/4).

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A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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A118395 Expansion of e.g.f. exp(x + x^3).

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A080893 Expansion of e.g.f. exp(xC(x)) = exp((1-sqrt(1-4x))/2), where C(x) is the g.f. of the Catalan numbers A000108.