A202829 -id:A202829 - cp's OEIS frontend

A202828 Expansion of e.g.f.: exp(4x/(1-2x)) / sqrt(1-4*x^2).

1, 4, 36, 400, 5776, 97344, 1915456, 42406144, 1049760000, 28558296064, 848579961856, 27271456395264, 943132599095296, 34877026635366400, 1373536895379849216, 57351382681767706624, 2530646978003730497536, 117614221470591038521344, 5742190572014854792806400

Offset: 0

Paul D. Hanna, Dec 25 2011

a(n) is the number of good involutions of the linear Alexander quandle (Z/4nZ, 2n+1); see Ta, Thm. 5.11 and cf. A387317. - Luc Ta, Aug 26 2025

			E.g.f.: A(x) = 1 + 4*x + 36*x^2/3! + 400*x^3/3! + 5776*x^4/4! + 97344*x^5/5! +...
where A(x) = 1 + 2^2*x + 6^2*x^2/2! + 20^2*x^3/3! + 76^2*x^4/4! + 312^2*x^5/5! +...+ A000898(n)^2*x^n/n! +...

G. C. Greubel, Table of n, a(n) for n = 0..350
Lực Ta, Good involutions of twisted conjugation subquandles and Alexander quandles, arXiv:2508.16772 [math.GT], 2025.

Cf. A000898, A202827, A202829, A202831, A202833, A202835, A202836.

Cf. A387317.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022

CoefficientList[Series[Exp[4*x/(1-2*x)]/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(4*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-2*k)*n!/((n-2*k)!*k!))^2}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)*2^k)^2}

def A202828_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202828_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A000898(n)^2, where the e.g.f. of A000898 is exp(2*x + x^2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-2*k) * n!/((n-2*k)!*k!) )^2.
a(n) = ( Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k) )^2. [From formula by Vladeta Jovovic in A000898].
a(n) ~ n^n*exp(2*sqrt(2*n)-1-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = 2*(n+1)*a(n-1) + 4*(n-1)*(n+1)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013
a(n) = 2^n*A277378(n). - R. J. Mathar, Jan 20 2020

A202831 Expansion of e.g.f.: exp(4x/(1-5x)) / sqrt(1-25*x^2).

Original entry on oeis.org

1, 4, 81, 1444, 44521, 1397124, 58354321, 2574344644, 136043683281, 7657406908804, 489836445798001, 33351743794661604, 2504378700538997881, 199445618093659242244, 17189578072429077875121, 1564487078400498014277124, 152146464623361858013314721

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: 1 + 4*x + 81*x^2/2! + 1444*x^3/3! + 44521*x^4/4! + 1397124*x^5/5! + ...
where A(x) = 1 + 2^2*x + 9^2*x^2/2! + 38^2*x^3/3! + 211^2*x^4/4! + 1182^2*x^5/5! + ... + A202832(n)^2*x^n/n! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A202832, A202827, A202828, A202829, A202833, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-5*x))/Sqrt(1-25*x^2) ))); // G. C. Greubel, Jun 21 2022

CoefficientList[Series[Exp[4*x/(1-5*x)]/Sqrt[1-25*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(4*x/(1-5*x)+x*O(x^n))/sqrt(1-25*x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)^2}

{a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))^2}

def A202831_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-5*x))/sqrt(1-25*x^2) ).egf_to_ogf().list()
A202831_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A202832(n)^2, where the e.g.f. of A202832 is exp(2*x + 5*x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(4*sqrt(n/5)-2/5-n)*5^n/2. - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (5*n-1)*a(n-1) + 5*(n-1)*(5*n-1)*a(n-2) - 125*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202836 Expansion of e.g.f.: exp(9x/(1-4x)) / sqrt(1-16*x^2).

Original entry on oeis.org

1, 9, 169, 3969, 119025, 4173849, 169754841, 7764958161, 395853630561, 22158814509225, 1352182116776841, 89167147951863969, 6319166996322943569, 478498255838869322169, 38549853656690487255225, 3290600595687160597292529, 296613603422471046790496961

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 9*x + 169*x^2/2! + 3969*x^3/3! + 119025*x^4/4! + ...
where A(x) = 1 + 3^2*x + 13^2*x^2/2! + 63^2*x^3/3! + 345^2*x^4/4! + 2043^2*x^5/5! + ... + A202837(n)^2*x^n/n! + ...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A202837, A202827, A202828, A202829, A202831, A202833, A202835.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-4*x))/Sqrt(1-16*x^2) ))); // G. C. Greubel, Jun 22 2022

CoefficientList[Series[Exp[9*x/(1-4*x)]/Sqrt[1-16*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(9*x/(1-4*x)+x*O(x^n))/sqrt(1-16*x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(3*x+2*x^2+x*O(x^n)),n)^2}

{a(n)=sum(k=0,n\2,3^(n-2*k)*2^k*n!/((n-2*k)!*k!))^2}

def A202836_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-4*x))/sqrt(1-16*x^2) ).egf_to_ogf().list()
A202836_list(40) # G. C. Greubel, Jun 22 2022

a(n) = A202837(n)^2, where the e.g.f. of A202837 is exp(3*x + 2*x^2).
a(n) ~ n^n*exp(3*sqrt(n)-9/8-n)*2^(2*n-1) * (1+33/(32*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (4*n+5)*a(n-1) + 4*(n-1)*(4*n+5)*a(n-2) - 64*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202827 Expansion of e.g.f.: exp(4*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 4, 25, 196, 1849, 20164, 249001, 3422500, 51739249, 851822596, 15155825881, 289527934084, 5906625426025, 128089110981316, 2940882813228649, 71239270847432164, 1815115761586307041, 48511703775281296900, 1356708799439194070809, 39615996090901693902916

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 4*x + 25*x^2/2! + 196*x^3/3! + 1849*x^4/4! +...
where A(x) = 1 + 2^2*x + 5^2*x^2/2! + 14^2*x^3/3! + 43^2*x^4/4! +...+ A005425(n)^2*x^n/n! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A005425, A202828, A202829, A202831, A202833, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 21 2022

With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-x)]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Dec 31 2011 *)

{a(n)=n!*polcoeff(exp(4*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-3*k)*n!/((n-2*k)!*k!))^2}

def A202827_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202827_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A005425(n)^2, where the e.g.f. of A005425 is exp(2*x + x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*n!/((n-2*k)!*k!) )^2. [From formula by Huajun Huang in A005425]
a(n) ~ n^n*exp(4*sqrt(n)-2-n)/2 * (1+5/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+3)*a(n-1) +(n-1)*(n+3)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 9, 100, 1296, 19044, 311364, 5588496, 108993600, 2291345424, 51585311376, 1236953249856, 31447331115264, 844332494760000, 23859653712215616, 707522071322329344, 21958125453144843264, 711555574637600891136, 24025060090437573945600

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 9*x + 100*x^2/2! + 1296*x^3/3! + 19044*x^4/4! + ...
where A(x) = 1 + 3^2*x + 10^2*x^2/2! + 36^2*x^3/3! + 138^2*x^4/4! + ... + A202834(n)^2*x^n/n! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A202834, A202827, A202828, A202829, A202831, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(9*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022

With[{nn=20},CoefficientList[Series[Exp[((9x)/(1-x))]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Apr 23 2012 *)

{a(n)=n!*polcoeff(exp(9*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(3*x+x^2/2+x*O(x^n)),n)^2}

def A202833_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202833_list(40) # G. C. Greubel, Jun 22 2022

a(n) = A202834(n)^2, where the e.g.f. of A202834 is exp(3*x + x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(6*sqrt(n)-9/2-n)/2 * (1+15/(4*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+8)*a(n-1) + (n-1)*(n+8)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202835 Expansion of e.g.f.: exp(9x/(1-2x)) / sqrt(1-4*x^2).

Original entry on oeis.org

1, 9, 121, 2025, 40401, 927369, 24000201, 689220009, 21710549025, 743187098889, 27441452694681, 1086166287819369, 45846179189949681, 2054407698719865225, 97357866191666622441, 4862830945258077841449, 255239441235423753980481, 14040944744510973314880009

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 9*x + 121*x^2/2! + 2025*x^3/3! + 40401*x^4/4! +...
where A(x) = 1 + 3^2*x + 11^2*x^2/2! + 45^2*x^3/3! + 201^2*x^4/4! + 963^2*x^5/5! +...+ A083886(n)^2*x^n/n! +...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A083886, A202827, A202828, A202829, A202831, A202833, A202836.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022

CoefficientList[Series[Exp[9*x/(1-2*x)]/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(9*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(3*x+x^2+x*O(x^n)),n)^2}

{a(n)=sum(k=0,n\2,3^(n-2*k)*n!/((n-2*k)!*k!))^2}

def A202835_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202835_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A083886(n)^2, where the e.g.f. of A083886 is exp(3*x + x^2).
a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k) * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(3*sqrt(2*n)-9/4-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (2*n+7)*a(n-1) + 2*(n-1)*(2*n+7)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202830 E.g.f.: exp(2x + 3x^2/2).

Original entry on oeis.org

1, 2, 7, 26, 115, 542, 2809, 15374, 89737, 548450, 3519799, 23493098, 163139563, 1172030654, 8706504265, 66638295998, 525069283921, 4248776775746, 35276087031463, 299986119953210, 2610709200699811, 23220585598592222, 210915850841272537, 1954390351189631726

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

From Emanuele Munarini, May 16 2014: (Start)
a(n) is the number of partitions of an n-set where each block consists of one or two elements, and each block is endowed with a permutation or a linear order.
For instance, for n = 2, we have the following partitions of the set {1,2}:
  (1)(2), (1)[2], [1](2), [1][2], (12), [12], [21] where parenthesis denote blocks consisting of cycles, and square brackets denote blocks consisting of linear orders. (End)

			E.g.f.: A(x) = 1 + 2*x + 7*x^2/2! + 26*x^3/3! + 115*x^4/4! + 542*x^5/5! + ...

Column k=3 of A376826.

Cf. A202829.

a := n -> I^(1-n)*2^((1+n)/2)*3^((n-1)/2)*KummerU((1-n)/2, 3/2, -2/3);
seq(round(evalf(a(n), 32)), n=0..23); # Peter Luschny, May 15 2014

CoefficientList[Series[E^(2*x+3*x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := Sum[StirlingS1[n, k]*3^n/2^(n - k) BellB[k, 2/3], {k, 0, n}]; Table[a[n], {n, 0, 12}] (* Emanuele Munarini, May 15 2014 *)

B(n,x) := sum(stirling2(n,k)*x^k,k,0,n);
a(n) := sum(stirling1(n,k)*3^n/2^(n-k)*B(k,2/3),k,0,n);
makelist(a(n),n,0,40); /* Emanuele Munarini, May 15 2014 */

{a(n)=n!*polcoeff(exp(2*x+3*x^2/2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-3*k)*3^k*n!/((n-2*k)!*k!))}

/* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+2*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-2*x-3*(n-k)*x^2*CF)); polcoeff(CF, n)}

a(n) = Sum_{k=0..floor(n/2)} 2^(n-3*k)*3^k * n!/((n-2*k)!*k!).
O.g.f.: 1/(1-2*x - 3*x^2/(1-2*x - 6*x^2/(1-2*x - 9*x^2/(1-2*x - 12*x^2/(1-2*x -...))))), (continued fraction).
E.g.f.: exp((4*x+3*x^2)/2) = G(0); G(k) = 1+(4*x+3*x^2)/(4*k+2-(4*x+3*x^2)*(4*k+2)/(4*x+3*x^2+4*(k+1)/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 28 2011
a(n) = 2*a(n-1) + 3*(n-1)*a(n-2), a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 29 2012
a(n) ~ exp((2/3)*sqrt(3*n) - n/2 - 1/3)*3^(n/2)*n^(n/2)/sqrt(2)*(1 + (11/54)*sqrt(3)/sqrt(n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*(3^n/2^(n-k))*B(k,2/3), where the s(n,k) are the (signless) Stirling numbers of the first kind and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials (or exponential polynomials), where the S(n,k) are the Stirling numbers of the second kind. - Emanuele Munarini, May 15 2014
a(n) = i^(1-n)*2^((1+n)/2)*3^((n-1)/2)*U((1-n)/2, 3/2, -2/3), U Kummer's hypergeometric function of the second kind. - Peter Luschny, May 15 2014

A202878 Expansion of e.g.f.: exp(16*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 16, 289, 5776, 126025, 2972176, 75186241, 2027520784, 57988974481, 1751546371600, 55668326576641, 1855807478279056, 64713593898036889, 2354701531657512976, 89209297718289390625, 3512141211682081889296, 143435878498076017059361

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 16*x + 289*x^2/2! + 5776*x^3/3! + 126025*x^4/4! + ...
where A(x) = 1 + 4^2*x + 17^2*x^2/2! + 76^2*x^3/3! + 355^2*x^4/4! + 1724^2*x^5/5! + ... + A202879(n)^2*x^n/n! + ...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A202879, A202827, A202828, A202829, A202831, A202835, A202836, A202837.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(16*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022

CoefficientList[Series[Exp[16*x/(1-x)]/Sqrt[1-x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(16*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(4*x+x^2/2+x*O(x^n)),n)^2}

def A202878_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(16*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202878_list(40) # G. C. Greubel, Jun 22 2022

a(n) = A202879(n)^2, where the e.g.f. of A202879 is exp(4*x + x^2/2).
a(n) = ( Sum_{k=0..floor(n/2)} 4^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(8*sqrt(n)-8-n)/2 * (1+22/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+15)*a(n-1) + (n-1)*(n+15)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

cp's OEIS Frontend

A202828 Expansion of e.g.f.: exp(4*x/(1-2*x)) / sqrt(1-4*x^2).

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A202831 Expansion of e.g.f.: exp(4*x/(1-5*x)) / sqrt(1-25*x^2).

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A202836 Expansion of e.g.f.: exp(9*x/(1-4*x)) / sqrt(1-16*x^2).

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SageMath

Formula

A202827 Expansion of e.g.f.: exp(4*x/(1-x)) / sqrt(1-x^2).

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Magma

Mathematica

PARI

PARI

SageMath

Formula

A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).

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Magma

Mathematica

PARI

PARI

SageMath

Formula

A202835 Expansion of e.g.f.: exp(9*x/(1-2*x)) / sqrt(1-4*x^2).

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A202828 Expansion of e.g.f.: exp(4x/(1-2x)) / sqrt(1-4*x^2).

A202831 Expansion of e.g.f.: exp(4x/(1-5x)) / sqrt(1-25*x^2).

A202836 Expansion of e.g.f.: exp(9x/(1-4x)) / sqrt(1-16*x^2).

A202835 Expansion of e.g.f.: exp(9x/(1-2x)) / sqrt(1-4*x^2).

A202830 E.g.f.: exp(2x + 3x^2/2).