A000932 a(n) = a(n-1) + n*a(n-2); a(0) = a(1) = 1. (Formerly M2595 N1025)

1, 1, 3, 6, 18, 48, 156, 492, 1740, 6168, 23568, 91416, 374232, 1562640, 6801888, 30241488, 139071696, 653176992, 3156467520, 15566830368, 78696180768, 405599618496, 2136915595392, 11465706820800, 62751681110208, 349394351630208, 1980938060495616

Offset

0,3

Comments

From Gary W. Adamson, Apr 20 2009: (Start)

Uses the same recursive operation as A000085.

Eigensequence of an infinite lower triangular matrix with (1, 1, 1, ...) as the main diagonal and (0, 2, 3, 4, 5, ...) as the subdiagonal. To generate A000085, replace the "0" in the subdiagonal with "1". (End)

References

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).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..799 (terms 0..200 from T. D. Noe)

Michael J. Kearney and Richard J. Martin, A note on an absorption problem for a Brownian particle moving in a harmonic potential, arXiv:2104.03183 [cond-mat.stat-mech], 2021.

Formula

From Paul D. Hanna, Aug 23 2011: (Start)

E.g.f. satisfies: A(x) = 1 + (1+x)*Integral A(x) dx.

E.g.f. satisfies: A(x) = A'(x)/(1+x) - (A(x)-1)/(1+x)^2.

If offset 1, then e.g.f. A(x) satisfies: F(A(x)) = 1 + x, where F(x) equals the e.g.f. of A173895 and satisfies: F'(x) = 1/(1 + x*F(x)). (End)

a(n)/a(n-1) = sqrt(n)+1/2+o(1) - Benoit Cloitre, Jul 02 2004

a(n) = -sqrt(Pi)/2*Sum[(-1)^k*2^(k/2)*Binomial[n,k]*(HypergeometricPFQRegularized[{1,k-n},{1+(k-n)/2,(1/2)*(1+k-n)},-(1/2)]+(-k+n)*HypergeometricPFQRegularized[{1,1+k-n},{1+(k-n)/2,(1/2)*(3+k-n)},-(1/2)])*HypergeometricU[1-k/2,3/2,1/2],{k,1,n}]. - Eric W. Weisstein, May 08 2013

E.g.f.: (1/2)*(2+e^(1/2*(1+x)^2)*sqrt(2*Pi)*(1+x)*(-erf(1/sqrt(2))+erf((1+x)/sqrt(2)))). - Eric W. Weisstein, May 08 2013

a(n) ~ sqrt(Pi)*(1-erf(1/sqrt(2)))/2 * n^(n/2+1/2)*exp(sqrt(n)-n/2+1/4) * (1+19/(24*sqrt(n))). - Vaclav Kotesovec, Aug 10 2013

a(n) = Sum_{k=0..n} A180048(n,k). - Philippe Deléham, Oct 28 2013

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 6*x^3/3! + 18*x^4/4! + 48*x^5/5! + 156*x^6/6! + ...
If offset 1, then e.g.f. A(x) = x + x^2/2! + 3*x^3/3! + 6*x^4/4! + 18*x^5/5! + 48*x^6/6! + 156*x^7/7! + ... + a(n-1)*x^n/n! + ...
satisfies F(A(x)) = 1 + x, where F(x) = e.g.f. of A173895:
F(x) = 1 + x - x^2/2! + 9*x^4/4! - 48*x^5/5! + 15*x^6/6! + 2448*x^7/7! + ...

Mathematica

RecurrenceTable[{a[n] == a[n - 1] + n a[n - 2], a[0] == a[1] == 1}, a, {n, 26}] (* Eric W. Weisstein, May 08 2013 *)
t = {1, 1}; Do[AppendTo[t, t[[-1]] + n*t[[-2]]], {n, 2, 30}]; t (* T. D. Noe, Jun 21 2012 *)
f[x_]:=2^(-x/2-2)*Sqrt[Pi*E]*(Erf[1/Sqrt[2]]-1)*(HermiteH[x+1, I/Sqrt[2]]*(Sin[Pi*x/2]+I*Cos[Pi*x/2])+HermiteH[x+1, -I/Sqrt[2]]*(Sin[Pi*x/2]-I*Cos[Pi*x/2]))+2^(x/2+1)*Cos[Pi*x]*Gamma[x+2]*HermiteH[-x-2, 1/Sqrt[2]]
Expand[FunctionExpand[Array[f, 20, 0]]] (* Velin Yanev, Oct 13 2021 *)

Crossrefs

Cf. A173895, A000085.

Sequence in context: A287212 A083337 A019308 * A187124 A369530 A161006

Adjacent sequences: A000929 A000930 A000931 * A000933 A000934 A000935

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Benoit Cloitre, Jul 02 2004

Status

approved