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A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k).

%I A183160 #67 Aug 17 2025 02:58:04
%S A183160 1,2,11,62,367,2232,13820,86662,548591,3498146,22436251,144583496,
%T A183160 935394436,6071718512,39523955552,257913792342,1686627623151,
%U A183160 11050540084902,72522925038257,476669316338542,3137209052543927,20672732229560032,136374124374593072,900541325129687272
%N A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k).
%H A183160 Vincenzo Librandi, <a href="/A183160/b183160.txt">Table of n, a(n) for n = 0..200</a>
%F A183160 a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k).
%F A183160 Self-convolution of A183161 (an integer sequence):
%F A183160 a(n) = Sum_{k=0..n} A183161(k)*A183161(n-k).
%F A183160 a(n) = Sum_{k=0..n} binomial(2*n+k,k) * cos((n+k)*Pi). - _Arkadiusz Wesolowski_, Apr 02 2012
%F A183160 Recurrence: 320*n*(2*n-1)*a(n) = 8*(346*n^2 + 79*n - 327)*a(n-1) + 6*(1688*n^2-6241*n+5981)*a(n-2) + 261*(3*n-7)*(3*n-5)*a(n-3). - _Vaclav Kotesovec_, Oct 20 2012
%F A183160 a(n) ~ 3^(3*n+3/2)/(2^(2*n+3)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 20 2012
%F A183160 ...
%F A183160 G.f.: A(x) = 1/(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - _Paul D. Hanna_, Nov 03 2012
%F A183160 G.f.: A(x) = 1 + x*d/dx { log( G(x)^5/(1+x*G(x)^2) )/2 }, where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - _Paul D. Hanna_, Nov 04 2012
%F A183160 G.f.: A(x) = 1/(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - _Paul D. Hanna_, Jun 16 2013
%F A183160 a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],2). - _Peter Luschny_, May 19 2015
%F A183160 a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n)). - _Ilya Gutkovskiy_, Oct 25 2017
%F A183160 From _G. C. Greubel_, Feb 22 2021: (Start)
%F A183160 a(n) = Sum_{k=0..n} A171822(n, k).
%F A183160 a(n) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1). (End)
%F A183160 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-2*k-1,n-2*k). - _Seiichi Manyama_, Apr 05 2024
%F A183160 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k). - _Seiichi Manyama_, Aug 03 2025
%F A183160 From _Seiichi Manyama_, Aug 14 2025: (Start)
%F A183160 a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
%F A183160 G.f.: g^2/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)
%F A183160 G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A005809. - _Seiichi Manyama_, Aug 15 2025
%e A183160 G.f.: A(x) = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 +...
%e A183160 A(x)^(1/2) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...+ A183161(n)*x^n +...
%e A183160 Given triangle A085478(n,k) = C(n+k,n-k), which begins:
%e A183160   1;
%e A183160   1,  1;
%e A183160   1,  3,  1;
%e A183160   1,  6,  5,  1;
%e A183160   1, 10, 15,  7, 1;
%e A183160   1, 15, 35, 28, 9, 1; ...
%e A183160 ILLUSTRATE formula a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k):
%e A183160 a(2) = 11 = 1*1 + 3*3 + 1*1;
%e A183160 a(3) = 62 = 1*1 + 6*5 + 5*6 + 1*1;
%e A183160 a(4) = 367 = 1*1 + 10*7 + 15*15 + 7*10 + 1*1;
%e A183160 a(5) = 2232 = 1*1 + 15*9 + 35*28 + 28*35 + 9*15 + 1*1; ...
%t A183160 Table[Sum[Binomial[n+k,n-k]Binomial[2n-k,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Jul 19 2011 *)
%t A183160 Table[HypergeometricPFQ[{-n, -n, 1/2 -n, n+1}, {1/2, 1, -2*n}, 1], {n, 0, 25}] (* _G. C. Greubel_, Feb 22 2021 *)
%o A183160 (PARI) {a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(2*n-k,k))}
%o A183160 (PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1-2*x*G^2-3*x^2*G^4), n)} \\ _Paul D. Hanna_, Nov 03 2012
%o A183160 (PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1+3*x*G-5*x*G^2), n)} \\ _Paul D. Hanna_, Jun 16 2013
%o A183160 for(n=0, 30, print1(a(n), ", "))
%o A183160 (Sage)
%o A183160 a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],2)
%o A183160 [simplify(a(n)) for n in range(26)] # _Peter Luschny_, May 19 2015
%o A183160 (Magma) [(&+[Binomial(n+k, 2*k)*Binomial(2*n-k, k): k in [0..n]]): n in [0..25]]; // _G. C. Greubel_, Feb 22 2021
%Y A183160 Cf. A001764, A085478, A183161, A199033, A218622.
%Y A183160 Cf. A026641, A147855, A371753.
%Y A183160 Cf. A005809, A006256, A160906, A226751.
%Y A183160 Cf. A171822.
%K A183160 nonn
%O A183160 0,2
%A A183160 _Paul D. Hanna_, Dec 27 2010