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

Author

Paul D. Hanna, Jan 20 2006

Keywords

• nonn
• easy

Comments

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

Links

• 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.

Crossrefs

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).

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

SageMath

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

Formula

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

Extensions

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