A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2. (Formerly M1449 N0573)

1, 2, 5, 13, 38, 116, 382, 1310, 4748, 17848, 70076, 284252, 1195240, 5174768, 23103368, 105899656, 498656912, 2404850720, 11879332048, 59976346448, 309442319456, 1628921941312, 8746095288800, 47840221880288, 266492604100288, 1510338372987776

Offset

1,2

Comments

a(n) is the number of set partitions of [n] in which the block containing 1 is of length <= 3 and all other blocks are of length <= 2. Example: a(4)=13 counts all 15 partitions of [4] except 1234 and 1/234. - David Callan, Jul 22 2008

Empirical: a(n) is the sum of the entries in the second-last row of the lower-triangular matrix of coefficients giving the expansion of degree-(n+1) complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86 (divided by 2).

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

John Cerkan, Table of n, a(n) for n = 1..795

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

Formula

a(n) = (1/2)*A000085(n+1).

E.g.f.: (1/2)*( (1+x)*exp(x + x^2/2) - 1). - Vladeta Jovovic, Nov 04 2003

Given e.g.f. y(x), then 0 = y'(x) * (1+x) - (y(x)+1/2) * (2+2*x+x^2) = 1 - y''(x) + y'(x)*(1 + x) + 2*y(x). - Michael Somos, Jan 23 2018

0 = +a(n)*(+a(n+1) +a(n+2) -a(n+3)) +a(n+1)*(-a(n+1) +a(n+2)) for all n>0. - Michael Somos, Jan 23 2018

a(n) ~ n^((n+1)/2) / (2^(3/2) * exp(n/2 - sqrt(n) + 1/4)) * (1 + 19/(24*sqrt(n))). - Vaclav Kotesovec, Apr 01 2018

Example

G.f. = x + 2*x + 5*x^2 + 13*x^3 + 38*x^4 + 116*x^5 + 382*x^6 + 1310*x^7 + ... - Michael Somos, Jan 23 2018

Maple

a := proc(n) option remember: if n = 1 then 1 elif n = 2 then 2 elif  n >= 3 then procname(n-1) +n*procname(n-2) fi; end:
seq(a(n), n = 1..100); # Muniru A Asiru, Jan 25 2018

Mathematica

RecurrenceTable[{a[1]==1, a[2]==2, a[n]==a[n-1]+n a[n-2]}, a, {n, 30}] (* Harvey P. Dale, Apr 21 2012 *)
(* Programs from Michael Somos, Jan 23 2018 *)
a[n_]:= With[{m=n+1}, If[m<2, 0, Sum[(2 k-1)!! Binomial[m, 2 k], {k, 0, m/2}]/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricU[-m/2, 1/2, -1/2] / (-1/2)^(m/2)/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricPFQ[{-m/2, (1-m)/2}, {}, 2]/2]];
a[n_]:= If[ n<1, 0, n! SeriesCoefficient[Exp[x+x^2/2]*(1+x)/2, {x, 0, n}]]; (* End *)
Fold[Append[#1, #1[[-1]] + #2 #1[[-2]]] &, {1, 2}, Range[3, 26]] (* Michael De Vlieger, Jan 23 2018 *)

Prog

(PARI) {a(n) = if( n<1, 0, n! * polcoeff( exp( x + x^2/2 + x * O(x^n)) * (1 + x) / 2, n))}; /* Michael Somos, Jan 23 2018 */
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1/2)*( (1+x)*exp(x + x^2/2) - 1))) \\ Joerg Arndt, Sep 04 2023
(GAP) a:=[1, 2];; for n in [3..10^2] do a[n] := a[n-1] + n*a[n-2]; od; a;  # Muniru A Asiru, Jan 25 2018
(Magma) I:=[1, 2]; [n le 2 select I[n] else Self(n-1)+n*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 31 2018
(SageMath)
def A001475_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( ((1+x)*exp(x+x^2/2) -1)/2 ).egf_to_ogf().list()
a=A001475_list(40); a[1:] # G. C. Greubel, Sep 03 2023

Crossrefs

Cf. A000085, A001189, A013989, A076276, A248475.

Sequence in context: A064384 A148302 A149857 * A360271 A343937 A369729

Adjacent sequences: A001472 A001473 A001474 * A001476 A001477 A001478

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Apr 21 2012

Status

approved