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A012244 a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 1, a(1) = 1. 11
1, 1, 4, 24, 204, 2220, 29520, 463680, 8401680, 172504080, 3958113600, 100370793600, 2787459998400, 84139894238400, 2742857884166400, 96034297911552000, 3594206259195552000, 143193586818810528000, 6050501147565883008000, 270263264589232282368000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of n-letter words from an n-letter alphabet such that no letter appears more than twice. - Paul Boddington, Nov 17 2003

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..393

K. S. Brown, Integer Sequences Related To PI [broken link]

D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.

FORMULA

E.g.f.: A(x) = (1 - 2*x - x^2)^(-1/2). - Paul Boddington, Nov 17 2003

a(n) = n!/2^n*A006139(n) = n!*Sum_{k=floor(n/2)..n} 2^(k-n)*C(n, k)*C(k, n-k). Sum_{n>=0} a(n)*x^n/n!^2 = exp(x)*BesselI(0, sqrt(2)*x). a(n) is the central coefficient of n!*(1+x+x^2/2)^n. - Vladeta Jovovic, Mar 22 2004

From Peter Bala, Aug 25 2011: (Start)

The function B(x) := int {t=0..x} A(t), obtained by integrating the generating function A(x), satisfies the autonomous differential equation d/dx(B(x)) = 1/(cos(B(x))-sin(B(x))). Compare with A190392.

Thus B(x), and hence A(x), can be found by inverting the function int {t=0..x} (cos(t)-sin(t)). By applying [Dominici, Theorem 4.1] the result can be expressed as

A(x) = 1 + sum {n>=1} D^n[1/(cos(t)-sin(t))](0)*x^n/n!, where the nested derivative D^n[f](x) of a function f(x) is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. Thus a(n) = D^n[1/(cos(t)-sin(t))](0). (End)

E.g.f. at offset 1: Series_Reversion(cos(x) + sin(x) - 1). - Paul D. Hanna, Aug 08 2012

a(n) ~ (1+sqrt(2))^(n+1/2) * n^n / (2^(1/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017

MAPLE

f := proc(n) option remember; if n <= 1 then 1 else (2*n-1)*f(n-1) +(n-1)^2*f(n-2); fi; end;

MATHEMATICA

Range[0, 20]! CoefficientList[Series[1/(1-2x-x^2)^(1/2), {x, 0, 20}], x]  (* Geoffrey Critzer, Dec 07 2011 *)

PROG

(PARI) {a(n)=local(X=x+x^2*O(x^n)); (n+1)!*polcoeff(serreverse(cos(X)+sin(X)-1), n+1)} \\ Paul D. Hanna, Aug 08 2012

CROSSREFS

Cf. A089975, A190392, A054765, A054766.

Sequence in context: A136229 A138419 A089946 * A240429 A240297 A050388

Adjacent sequences:  A012241 A012242 A012243 * A012245 A012246 A012247

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 19 11:06 EDT 2019. Contains 327192 sequences. (Running on oeis4.)