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A055834
a(n) = T(2n,n), where T is the array in A055830.
3
1, 1, 4, 18, 85, 413, 2044, 10248, 51876, 264550, 1357070, 6994780, 36196706, 187938842, 978599560, 5108177816, 26721644973, 140050505085, 735254208670, 3865837887450, 20353393741065, 107290306033845, 566194674179160, 2990958274811520, 15814562990604300, 83690040760923168
OFFSET
0,3
LINKS
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
FORMULA
a(n) = Sum_{k=0..n} binomial(n+k-1,n)*binomial(k,n-k). - Max Alekseyev, Jun 17 2007
Recurrence: 5*(n-1)*n*a(n) = 2*(n-1)*(11*n-3)*a(n-1) + 3*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 27^n/5^n*sqrt(2/(15*Pi*n)). - Vaclav Kotesovec, Nov 19 2012
a(n) = A055835(n)/3 for n>=1. - Philippe Deléham, Jan 25 2014
G.f.: x*B'(x)-x*B'(x)/B(x)+B(x), where B(x) is g.f. of A001002. - Vladimir Kruchinin, Sep 20 2015
MAPLE
seq( add(binomial(n+k-1, n)*binomial(k, n-k), k=0..n), n=0..30); # G. C. Greubel, Jan 21 2020
MATHEMATICA
Table[Sum[Binomial[n+k-1, n]Binomial[k, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Oct 03 2011 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+k-1, n)*binomial(k, n-k)); \\ Joerg Arndt, May 06 2013
(Maxima)
b(n):= sum(binomial(n+k, k)*binomial(k, n-k), k, ceiling(n/2), n)/(n+1);
B(x):= sum(b(i)*x^(i), i, 0, 30);
makelist(coeff(taylor(x*diff(B(x), x)-x*diff(B(x), x)/B(x)+B(x), x, 0, 20), x, n), n, 0, 20); /* Vladimir Kruchinin, Sep 21 2015 */
(Magma) [&+[(Binomial(n+k-1, n)*Binomial(k, n-k)): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Sep 21 2015
(Sage) [sum(binomial(n+k-1, n)*binomial(k, n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jan 21 2020
(GAP) List([0..30], n-> Sum([0..n], k-> Binomial(n+k-1, n)*Binomial(k, n-k)) ); # G. C. Greubel, Jan 21 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 28 2000
STATUS
approved