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 A116363 a(n) = dot product of row n in Catalan triangle A033184 with row n in Pascal's triangle. 2
 1, 2, 7, 30, 141, 698, 3571, 18686, 99385, 535122, 2908863, 15932766, 87809541, 486421770, 2706138987, 15110359038, 84637982961, 475381503266, 2676447372535, 15100548901790, 85357620588541, 483304834607322 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 D. Drake, Bijections from Weighted Dyck Paths to SchrÃ¶der Paths, J. Int. Seq. 13 (2010) # 10.9.2. FORMULA a(n) = Sum_{k=0..n} C(n,k)*C(2*n-k+1,n-k)*(k+1)/(2*n-k+1). G.f. A(x) satisfies: d/dx[log(1 - 4*x*A(x))] = -4*(1-5*x)/(1-13*x+43*x^2-7*x^3). O.g.f.: 2*(R+x)/(R*(R+x+1)), where R = sqrt(x^2+6*x+1). [Dan Drake, May 19 2010] Conjecture: +(2*n+5)*(n+1)*a(n) +4*(-3*n^2-9*n+5)*a(n-1) +(2*n+7)*(n-1)*a(n-2)=0. - R. J. Mathar, Jun 22 2016 EXAMPLE The dot product of Catalan row 4 and Pascal row 4 equals a(4) = [14,14,9,4,1]*[1,4,6,4,1] = 141 which is equivalent to obtaining the final term in these repeated partial sums of Pascal row 4: 1,4, 6, 4, 1 .5,11,15,16 ..16,31,47 ...47,94 ....141 MATHEMATICA Table[Sum[Binomial[n, j]*Binomial[2*n-j+1, n-j]*(j+1)/(2*n-j+1), {j, 0, n} ], {n, 0, 30}] (* G. C. Greubel, May 12 2019 *) PROG (PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(2*n-k+1, n-k)*(k+1)/(2*n-k+1)) for(n=0, 30, print1(a(n), ", ")) (MAGMA) [(&+[Binomial(n, j)*Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1): j in [0..n]]): n in [0..30]]; // G. C. Greubel, May 12 2019 (Sage) [sum(binomial(n, j)*binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1) for j in (0..n)) for n in (0..30)] # G. C. Greubel, May 12 2019 (GAP) List([0..30], n-> Sum([0..n], j-> Binomial(n, j)*Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1))) # G. C. Greubel, May 12 2019 CROSSREFS Cf. A033184. Sequence in context: A097924 A027136 A299296 * A186858 A260771 A260773 Adjacent sequences:  A116360 A116361 A116362 * A116364 A116365 A116366 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 04 2006 STATUS approved

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Last modified July 23 00:43 EDT 2019. Contains 325228 sequences. (Running on oeis4.)