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 A128196 A weighted sum of quotients of double factorials. 4
 1, 3, 13, 73, 527, 4775, 52589, 683785, 10257031, 174370039, 3313031765, 69573669113, 1600194393695, 40004859850567, 1080131215981693 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is the sum of rows in the following triangle (A126063): T(n,k) (n,k>=0) ...........1. ...........1,.......2 ...........3,.......6,.......4. ..........15,......30,......20,.......8 .........105,.....210,.....140,......56,.....16 .........945,....1890,....1260,.....504,....144,....32 .......10395,...20790,...13860,....5544,...1584,...352,....64 ......135135,..270270,..180180,...72072,..20592,..4576,...832,..128 First column is A001147, second column is A097801. The diagonal is A000079, the subdiagonal is A014480. Let H be the diagonal matrix diag(1,2,4,8,...) and let G be the matrix (n!! defined as A001147(n), -1!! = 1): (-1)!!/(-1)!! 1!!/(-1)!! 1!!/1!! 3!!/(-1)!! 3!!/1!! 3!!/3!! 5!!/(-1)!! 5!!/1!! 5!!/3!! 5!!/5!! ... Then T = G*H. [Gottfried Helms] LINKS P. Luschny, Variants of Variations. FORMULA a(n) = (2n)!/(n! 2^n) Sum(k=0..n, 4^k k!/(2k)!) a(n) = 2^n Gamma(n+1/2) Sum(k=0..n, 1/Gamma(k+1/2)) a(n) = Sum(k=0..n, 2^k n!!/k!!) [n!! defined as A001147(n), Gottfried Helms] a(n) = Sum(k=0..n, 2^(2k-n)((n+1)! Catalan(n))/((k+1)! Catalan(k))) [Catalan(n) A000108] a(n) = Sum(k=0..n, 2^(2k-n) QuadFact(n)/QuadFact(k)) [QuadFact(n) A001813] a(n) = Sum(k=0..n, 2^(2k-n) (-1)^(n-k) A097388(n)/A097388(k) ) a(n) = A001147(n) Sum(k=0..n, 2^k / A001147(k)) a(n) = A128195(n)/A005408(n) a(n) = A128195(n-1)+A000079(n) (if n>0) Recursive form: a(n) = (2n-1)*a(n-1) + 2^n; a(0) = 1 [Gottfried Helms] Note: The following constants will be used in the next formulas. K = (1-exp(1)*Gamma(1/2,1))/Gamma(1/2) M = sqrt(2)(1+exp(1)(Gamma(1/2)-Gamma(1/2,1))) Generalized form: For x>0 a(x) = 2^x(exp(1)*Gamma(x+1/2,1) + K*Gamma(x+1/2)) Asymptotic formula: a(n) ~ 2^n*(1+(exp(1)+K)*(n-1/2)!) a(n) ~ M(2exp(-1)(n-1/(24*n+19/10*1/n)))^n MAPLE a := n -> `if`(n=0, 1, (2*n-1)*a(n-1)+2^n); MATHEMATICA a[n_] := Sum[2^k*((2*n-1)!!/(2*k-1)!!), {k, 0, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 28 2013 *) CROSSREFS Cf. A128195, A001147, A126063. Sequence in context: A059294 A124468 A205572 * A162161 A119013 A190878 Adjacent sequences:  A128193 A128194 A128195 * A128197 A128198 A128199 KEYWORD easy,nonn AUTHOR Peter Luschny, Feb 26 2007 STATUS approved

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Last modified May 10 04:49 EDT 2021. Contains 343748 sequences. (Running on oeis4.)