A119259 - OEIS

%I #72 Dec 08 2024 06:54:51

%S 1,3,17,111,769,5503,40193,297727,2228225,16807935,127574017,

%T 973168639,7454392321,57298911231,441739706369,3414246490111,

%U 26447737520129,205272288591871,1595964714385409,12427568655368191,96905907580960769,756583504975757311,5913649000782757889

%N Central terms of the triangle in A119258.

%C The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - _Peter Bala_, Jan 06 2022

%D R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

%H Vincenzo Librandi, <a href="/A119259/b119259.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.2.6, eq (42).

%F a(n) = A119258(2*n,n).

%F a(n) = Sum_{k=0..n} C(2*n,k)*C(2*n-k-1,n-k). - _Paul Barry_, Sep 28 2007

%F a(n) = Sum_{k=0..n} C(n+k-1,k)*2^k. - _Paul Barry_, Sep 28 2007

%F 2*a(n) = A064062(n)+A178792(n). - _Joseph Abate_, Jul 21 2010

%F G.f.: (4*x^2+3*sqrt(1-8*x)*x-5*x)/(sqrt(1-8*x)*(2*x^2+x-1)-8*x^2-7*x+1). - _Vladimir Kruchinin_, Aug 19 2013

%F a(n) = (-1)^n - 2^(n+1)*binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2],2). - _Peter Luschny_, Jul 25 2014

%F a(n) = (-1)^n + 2^(n+1)*A014300(n). - _Peter Luschny_, Jul 25 2014

%F a(n) = [x^n] ( (1 + x)^2/(1 - x) )^n. Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 13*x^2 + 67*x^3 + ... is essentially the o.g.f. for A064062. - _Peter Bala_, Oct 01 2015

%F The o.g.f. is the diagonal of the bivariate rational function 1/(1 - t*(1 + x)^2/(1 - x)) and hence is algebraic by Stanley 1999, Theorem 6.33, p.197. - _Peter Bala_, Aug 21 2016

%F n*(3*n-4)*a(n) +(-21*n^2+40*n-12)*a(n-1) -4*(3*n-1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Aug 09 2017

%F From _Peter Bala_, Mar 23 2020: (Start)

%F a(p) == 3 ( mod p^3 ) for prime p >= 5. Cf. A002003, A103885 and A156894.

%F More generally, we conjecture that a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. (End)

%F G.f.: (8*x)/(sqrt(1-8*x)*(1+4*x)-1+8*x). - _Fabian Pereyra_, Jul 20 2024

%F a(n) = 2^(n+1)*binomial(2*n,n) - A178792(n). - _Akiva Weinberger_, Dec 06 2024

%t Table[Binomial[2k - 1, k] Hypergeometric2F1[-2k, -k, 1 - 2k, -1], {k, 0, 10}] (* _Vladimir Reshetnikov_, Feb 16 2011 *)

%o (Haskell)

%o a119259 n = a119258 (2 * n) n  -- _Reinhard Zumkeller_, Aug 06 2014

%o (Python)

%o from itertools import count, islice

%o def A119259_gen(): # generator of terms

%o     yield from (1,3)

%o     a, c = 2, 1

%o     for n in count(1):

%o         yield (a<<n+2)+(1 if n&1 else -1)

%o         a=(3*n+5)*(c:=c*((n<<2)+2)//(n+2))-a>>1

%o A119259_list = list(islice(A119259_gen(),20)) # _Chai Wah Wu_, Apr 26 2023

%Y Cf. A005408, A056220, A000225, A000337, A055580, A027608, A119258, A064062, A178792, A014300, A098665.

%K nonn,easy

%O 0,2

%A _Reinhard Zumkeller_, May 11 2006