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 A002737 a(n) = Sum_{j=0..n} (n+j)*binomial(n+j,j). (Formerly M3975 N1644) 2
 0, 5, 35, 189, 924, 4290, 19305, 85085, 369512, 1587222, 6760390, 28601650, 120349800, 504131940, 2103781365, 8751023325, 36300541200, 150217371150, 620309379690, 2556724903590, 10520494818600, 43225511319900, 177361820257050, 726860987017074, 2975511197688624, 12168371410300700 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The former title was "Coefficients for extrapolation". REFERENCES J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy]. J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages) FORMULA a(n) = Sum_{j=0..n} binomial(n+j,j)*(n+j). - Zerinvary Lajos, Aug 30 2006 a(n) = n*binomial(2*n+4, n+2)/4. - Zerinvary Lajos, Feb 28 2007 These 2 formulas are correct - see A331432. - N. J. A. Sloane, Jan 17 2020 a(n) = (n*(2*n + 3)*binomial(2*n + 1, n + 1))/(n + 2). - Peter Luschny, Jan 18 2020 E.g.f.: exp(2*x) * ((1 - 3*x + 8*x^2) * BesselI(1,2*x) / x - (1 - 8*x) * BesselI(0,2*x)). - Ilya Gutkovskiy, Nov 03 2021 G.f.: ((1-3*x -4*x^2)*sqrt(1-4*x) -(1-5*x))/(2*x^2*(1-4*x)^(3/2)). - G. C. Greubel, Mar 23 2022 MAPLE t5 := n-> add(binomial(n+j, j)*(n+j), j=0..n); [seq(t5(n), n=0..40)]; # Alternative: A002737 := n -> (n*(2*n + 3)*binomial(2*n+1, n+1))/(n + 2): seq(A002737(n), n=0..25); # Peter Luschny, Jan 18 2020 MATHEMATICA Table[n(2n+3)Binomial[2n+1, n+1]/(n+2), {n, 0, 25}] (* Vincenzo Librandi, Jan 19 2020 *) PROG (Magma) [(n*(2*n+3)*Binomial(2*n+1, n+1))/(n+2): n in [0..30]]; // Vincenzo Librandi, Jan 19 2020 (SageMath) [n*(n+3)*catalan_number(n+2)/4 for n in (0..30)] # G. C. Greubel, Mar 23 2022 CROSSREFS A diagonal of A331432. Cf. A000108. Sequence in context: A043014 A165755 A166149 * A241779 A265976 A123008 Adjacent sequences: A002734 A002735 A002736 * A002738 A002739 A002740 KEYWORD nonn AUTHOR N. J. A. Sloane EXTENSIONS Entry revised by N. J. A. Sloane, Jan 18 2020 STATUS approved

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Last modified September 18 05:14 EDT 2024. Contains 375995 sequences. (Running on oeis4.)