A173895 -id:A173895 - cp's OEIS frontend

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

N. J. A. Sloane

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)

			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! + ...

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).

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.

Cf. A173895, A000085.

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 *)

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

More terms from Benoit Cloitre, Jul 02 2004

A144010 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x)) with A(0)=1.

Original entry on oeis.org

1, 1, 1, 4, 21, 160, 1525, 17760, 243145, 3833600, 68373225, 1361264000, 29925477725, 719991897600, 18817847565725, 530921477363200, 16082605690148625, 520603130117939200, 17934634668874889425

Offset: 0

Paul D. Hanna, Sep 10 2008

nonn

From Peter Bala, Nov 26 2010: (Start)
Define a polynomial sequence P_n(x) recursively by
... P_0(x) = 1,
... P_n(x) = (x-1)*P_(n-1)(x-1) + n*P_(n-1)(x+1) for n >= 1.
The first few polynomials are
P_1(x) = x;
P_2(x) = x^2 + 3;
P_3(x) = x^3 + 12*x + 8.
It appears that a(n+1) = P_n(1) (checked as far as a(19)).
Compare with A173895. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A144011, A238302.

FindRoot[Sqrt[Pi/2]*s*E^(-s^2/2)*(Erfi[1/Sqrt[2]]-Erfi[s/Sqrt[2]]) == -1,{s,1},WorkingPrecision->50] (* program for numerical value of the constant s, Vaclav Kotesovec, Feb 23 2014 *)
a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[k Binomial[n-1, k] a[k] a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 17 2016 *)

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1-x*A+x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

/* from Peter Bala's Formula */
{a(n)=local(P=1);if(n>=0&n<2,1,for(k=1,n-1,P=(x-1)*subst(P,x,x-1) + k*subst(P,x,x+1)));subst(P,x,1)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Jun 15 2013

E.g.f. satisfies: A(x) = 1 + Integral 1/(1 - x*A(x)) dx.
a(n) ~ n^(n-1) * s^n / exp(n), where s = 2.0832144900084392272885741721727173082215... is the root of the equation sqrt(Pi/2)*s*exp(-s^2/2)*(erfi(1/sqrt(2)) - erfi(s/sqrt(2))) = -1. - Vaclav Kotesovec, Feb 23 2014
a(0) = 1, a(1) = 1, a(n) = Sum_{0 < k < n} k * binomial(n-1, k) * a(k) * a(n-k-1). - Vladimir Reshetnikov, May 17 2016

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions