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A065866 a(n) = n! * Catalan(n+1). 4
1, 2, 10, 84, 1008, 15840, 308880, 7207200, 196035840, 6094932480, 213322636800, 8303173401600, 355850288640000, 16653793508352000, 845180020548864000, 46236318771202560000, 2712530701243883520000, 169890080762116915200000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

R. L. Graham, D. E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, pp. 200-204.

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..100

FORMULA

a(n) = 2 * (2n+1)!/(n+2)!. E.g.f.: (1-2x-sqrt(1-4x))/(2x^2) = (O.g.f. for A000108)^2 = B_2(x)^2 (cf. GKP reference)

0 = a(n)*(-7200*a(n+2) + 2700*a(n+3) + 246*a(n+4) - 33*a(n+5)) + a(n+1)*(+36*a(n+2) + 372*a(n+3) + 36*a(n+4) - a(n+5)) + a(n+2)*(-18*a(n+2) + 9*a(n+3) + a(n+4)) for n>=0. - Michael Somos, Apr 14 2015

The e.g.f. A(x) satisfies 0 = -2 + A(x) * (6*x - 2) + A'(x) * (4*x^2 - x). - Michael Somos, Apr 14 2015

Conjecture: (n+2)*a(n) -2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Oct 31 2015

EXAMPLE

G.f. = 1 + 2*x + 10*x^2 + 84*x^3 + 1008*x^4 + 15840*x^5 + 308880*x^6 + ...

MAPLE

with(combstruct): ZL:=[T, {T=Union(Z, Prod(Epsilon, Z, T), Prod(T, Z, Epsilon), Prod(T, T, Z))}, labeled]: seq(count(ZL, size=i+1)/(i+1), i=0..18); # Zerinvary Lajos, Dec 16 2007

PROG

(PARI) { for (n = 0, 100, a = 2 * (2*n + 1)!/(n + 2)!; write("b065866.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

CROSSREFS

Cf. A000108.

Equals 2 * A102693(n+1), n>0.

Main diagonal of A256116.

Sequence in context: A113332 A180715 A107863 * A156466 A132397 A202745

Adjacent sequences:  A065863 A065864 A065865 * A065867 A065868 A065869

KEYWORD

nonn

AUTHOR

Len Smiley, Dec 06 2001

STATUS

approved

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Last modified December 10 11:36 EST 2016. Contains 279001 sequences.