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 A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k). 12
 1, 2, 11, 62, 367, 2232, 13820, 86662, 548591, 3498146, 22436251, 144583496, 935394436, 6071718512, 39523955552, 257913792342, 1686627623151, 11050540084902, 72522925038257, 476669316338542, 3137209052543927, 20672732229560032, 136374124374593072, 900541325129687272 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k). Self-convolution of A183161 (an integer sequence): a(n) = Sum_{k=0..n} A183161(k)*A183161(n-k). a(n) = sum(k=0..n, binomial(2*n+k,k)*cos((n+k)*Pi)). [Arkadiusz Wesolowski, Apr 02 2012] 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 a(n) ~ 3^(3*n+3/2)/(2^(2*n+3)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012 ... 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 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 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 a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],2). - Peter Luschny, May 19 2015 a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n)). - Ilya Gutkovskiy, Oct 25 2017 From G. C. Greubel, Feb 22 2021: (Start) a(n) = Sum_{k=0..n} A171822(n, k). a(n) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1). (End) EXAMPLE G.f.: A(x) = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 +... 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 +... Given triangle A085478(n,k) = C(n+k,n-k), which begins:   1;   1,  1;   1,  3,  1;   1,  6,  5,  1;   1, 10, 15,  7, 1;   1, 15, 35, 28, 9, 1; ... ILLUSTRATE formula a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k): a(2) = 11 = 1*1 + 3*3 + 1*1; a(3) = 62 = 1*1 + 6*5 + 5*6 + 1*1; a(4) = 367 = 1*1 + 10*7 + 15*15 + 7*10 + 1*1; a(5) = 2232 = 1*1 + 15*9 + 35*28 + 28*35 + 9*15 + 1*1; ... MATHEMATICA Table[Sum[Binomial[n+k, n-k]Binomial[2n-k, k], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Jul 19 2011 *) Table[HypergeometricPFQ[{-n, -n, 1/2 -n, n+1}, {1/2, 1, -2*n}, 1], {n, 0, 25}] (* G. C. Greubel, Feb 22 2021 *) PROG (PARI) {a(n)=sum(k=0, n, binomial(n+k, n-k)*binomial(2*n-k, k))} (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 (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 for(n=0, 30, print1(a(n), ", ")) (Sage) a = lambda n: binomial(3*n+1, n)*hypergeometric([1, -n], [2*n+2], 2) [simplify(a(n)) for n in range(26)] # Peter Luschny, May 19 2015 (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 CROSSREFS Cf. A001764, A085478, A183161, A199033, A218622. Cf. A171822. Sequence in context: A034726 A256933 A162274 * A020078 A002629 A235937 Adjacent sequences:  A183157 A183158 A183159 * A183161 A183162 A183163 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 27 2010 STATUS approved

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Last modified June 19 09:39 EDT 2021. Contains 345126 sequences. (Running on oeis4.)