A000932 a(n) = a(n-1) + n*a(n-2); a(0) = a(1) = 1.
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
Examples
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! + ...
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.
Programs
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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 *)
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
Extensions
More terms from Benoit Cloitre, Jul 02 2004
Comments