A000932 - OEIS

%I M2595 N1025 #84 Aug 21 2022 06:35:42

%S 1,1,3,6,18,48,156,492,1740,6168,23568,91416,374232,1562640,6801888,

%T 30241488,139071696,653176992,3156467520,15566830368,78696180768,

%U 405599618496,2136915595392,11465706820800,62751681110208,349394351630208,1980938060495616

%N a(n) = a(n-1) + n*a(n-2); a(0) = a(1) = 1.

%C From _Gary W. Adamson_, Apr 20 2009: (Start)

%C Uses the same recursive operation as A000085.

%C 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)

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A000932/b000932.txt">Table of n, a(n) for n = 0..799</a> (terms 0..200 from T. D. Noe)

%H Michael J. Kearney and Richard J. Martin, <a href="https://arxiv.org/abs/2104.03183">A note on an absorption problem for a Brownian particle moving in a harmonic potential</a>, arXiv:2104.03183 [cond-mat.stat-mech], 2021.

%F From _Paul D. Hanna_, Aug 23 2011: (Start)

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

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

%F 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)

%F a(n)/a(n-1) = sqrt(n)+1/2+o(1) - _Benoit Cloitre_, Jul 02 2004

%F 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

%F 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

%F 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

%F a(n) = Sum_{k=0..n} A180048(n,k). - _Philippe Deléham_, Oct 28 2013

%e 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! + ...

%e 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! + ...

%e satisfies F(A(x)) = 1 + x, where F(x) = e.g.f. of A173895:

%e F(x) = 1 + x - x^2/2! + 9*x^4/4! - 48*x^5/5! + 15*x^6/6! + 2448*x^7/7! + ...

%t 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 t = {1, 1}; Do[AppendTo[t, t[[-1]] + n*t[[-2]]], {n, 2, 30}]; t (* _T. D. Noe_, Jun 21 2012 *)

%t 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]]

%t Expand[FunctionExpand[Array[f,20,0]]] (* _Velin Yanev_, Oct 13 2021 *)

%Y Cf. A173895, A000085.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Benoit Cloitre_, Jul 02 2004