A220902 a(n) = Catalan(n) - A000245(n-2).
2, 4, 11, 33, 104, 339, 1133, 3861, 13364, 46852, 166022, 593674, 2139552, 7763305, 28337265, 103981965, 383351580, 1419269280, 5274495930, 19669409790, 73580417040, 276043317030, 1038327097314, 3915101867778, 14795310818024, 56028144245304, 212581753906508, 808027815817012
Offset: 2
Links
- Robert Israel, Table of n, a(n) for n = 2..1667
- Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012 (see b_n).
- F. Disanto and T. Wiehe, On the sub-permutations of pattern avoiding permutations, Discrete Math., 337 (2014), 127-141.
Programs
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Magma
[Catalan(n)-Catalan(n-1)+Catalan(n-2): n in [2..30]]; // Vincenzo Librandi, Dec 24 2015
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Maple
catalan:= n -> (2*n)!/n!/(n+1)!: A220902:= n -> catalan(n) - catalan(n-1)+catalan(n-2): map(A220902, [$2..100]); # Robert Israel, Dec 31 2015
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Mathematica
Table[CatalanNumber[n] - CatalanNumber[n-1] + CatalanNumber[n-2], {n, 2, 30}] (* Vincenzo Librandi, Dec 24 2015 *) CoefficientList[ Series[-x + (1 -Sqrt[1-4x])(1 -(-1 +Sqrt[1-4x])^3/(8x) +x)/2, {x, 0, 26}], x] (* Robert G. Wilson v, Dec 24 2015 *) -
PARI
my(x='x+O('x^50)); Vec((1+x+x^2*((1-sqrt(1-4*x))/(2*x))^3)*x*((1-sqrt(1-4*x))/(2*x))-x) \\ Altug Alkan, Dec 24 2015 -
Sage
[sum((-1)^j*catalan_number(n-j) for j in (0..2)) for n in (2..30)] # G. C. Greubel, May 03 2021
Formula
a(n) = Catalan(n) - Catalan(n-1) + Catalan(n-2). - Andrei Asinowski, Dec 16 2015
G.f.: (1 + x + x^2*C(x)^3)*x*C(x) - x where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 04 2014
Conjecture: (n+1)*a(n) +(-5*n+3)*a(n-1) +(5*n-13)*a(n-2) +2*(-2*n+9)*a(n-3)=0. - R. J. Mathar, May 30 2014
From Robert Israel, Dec 31 2015: (Start)
Mathar's conjecture can be verified by expressing a in terms of factorials and simplifying.
G.f.: (1-3*x+x^2 -(1-x+x^2)*sqrt(1-4*x))/(2*x). (End)
E.g.f.: (1/6)*(-3*(3-x) + exp(2*x) * ( (9 -15*x +8*x^2)*BesselI(0, 2*x) - (6 -13*x +8*x^2)*BesselI(1, 2*x) ) ). - G. C. Greubel, May 03 2021