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URL: https://oeis.org/A000932

⇱ A000932 - OEIS

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A000932
a(n) = a(n-1) + n*a(n-2); a(0) = a(1) = 1.
9
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)
Nathanaël Hassler and Sergey Kirgizov, Emerging consecutive pattern avoidance, arXiv:2511.02442 [math.CO], 2025. See Remark 4.9 p. 8.
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
KEYWORD
nonn,easy
EXTENSIONS
More terms from Benoit Cloitre, Jul 02 2004
STATUS
approved