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A086618 a(n) = Sum{k=0..n} binomial(n,k)^2*C(k), where C() = A000108() are the Catalan numbers. 11
1, 2, 7, 33, 183, 1118, 7281, 49626, 349999, 2535078, 18758265, 141254655, 1079364105, 8350678170, 65298467487, 515349097713, 4100346740511, 32858696386766, 265001681344569 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Main diagonal of square table A086617 of coefficients, T(n,k), of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^2.
a(n) is the number of permutations of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 4. - Eric S. Egge, Oct 21 2008
In 2012, Zhi-Wei Sun proved that for any odd prime p we have the congruence a(1) + ... + a(p-1) == 0 (mod p^2). - Zhi-Wei Sun, Aug 22 2013
LINKS
D. Daly and L. Pudwell, Pattern avoidance in rook monoids, 2013.
T. Denton, Algebraic and Affine Pattern Avoidance, arXiv preprint arXiv:1303.3767 [math.CO], 2013.
Z.-W. Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034 [math.NT], 2011. See (1.22).
Z.-W. Sun, On sums of Apery polynomials and related congruences, J. Number Theory 132(2012), 2673-2699.
FORMULA
Recurrence: (n+3)^2*(4*n+7)*a(n+2) = 2*(20*n^3+117*n^2+220*n+135)*a(n+1) - 9*(n+1)^2*(4*n+11)*a(n). - Vaclav Kotesovec, Sep 11 2012
a(n) ~ 3^(5/2)/(8*Pi) * 9^n/n^2. - Vaclav Kotesovec, Oct 06 2012
G.f.: (1-(1-9*x)^(1/3)*hypergeom([1/3,1/3],[1],-27*x*(1-x)^2/(1-9*x)^2))/(6*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([1/2,-n,-n], [1,2], 4). - Vladimir Reshetnikov, Oct 03 2016
D-finite with recurrence (n+1)^2*a(n) +(-19*n^2+8*n+6)*a(n-1) +9*(11*n^2-30*n+21)*a(n-2) -81*(n-2)^2*a(n-3)=0. - R. J. Mathar, Aug 01 2022
EXAMPLE
a(5) = 1118 = 1*1^2 + 1*5^2 + 2*10^2 + 5*10^2 + 14*5^2 + 42*1^2.
MATHEMATICA
Flatten[{1, RecurrenceTable[{(n+3)^2*(4*n+7)*a[n+2]==2*(20*n^3+117*n^2+220*n+135)*a[n+1]-9*(n+1)^2*(4*n+11)*a[n], a[1]==2, a[2]==7}, a, {n, 1, 20}]}] (* Vaclav Kotesovec, Sep 11 2012 *)
Table[HypergeometricPFQ[{1/2, -n, -n}, {1, 2}, 4], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)
PROG
(PARI) a(n)=sum(k=0, n-1, binomial(n-1, k)^2*binomial(2*k, k)/(k+1)) \\ Charles R Greathouse IV, Sep 12 2012
(PARI) a(n)=sum(k=0, n-1, (4*k+3)*sum(i=0, k, binomial(k, i)^2*binomial(2*i, i)))/3/n^2 \\ Charles R Greathouse IV, Sep 12 2012
CROSSREFS
Cf. A086617 (table), A086615 (antidiagonal sums), A003046 (determinants).
Cf. A000108.
Cf. A228456.
Sequence in context: A055724 A301433 A054727 * A224769 A302285 A249636
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 24 2003
EXTENSIONS
Edited by N. J. A. Sloane, Sep 14 2012. The formula in the new definition was first sent in by Michael Somos, Feb 19 2004
Minor edits Vaclav Kotesovec, Mar 31 2014
STATUS
approved

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Last modified March 19 06:25 EDT 2024. Contains 370953 sequences. (Running on oeis4.)