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%I M3975 N1644 #47 Mar 20 2024 15:40:24
%S 0,5,35,189,924,4290,19305,85085,369512,1587222,6760390,28601650,
%T 120349800,504131940,2103781365,8751023325,36300541200,150217371150,
%U 620309379690,2556724903590,10520494818600,43225511319900,177361820257050,726860987017074,2975511197688624,12168371410300700
%N a(n) = Sum_{j=0..n} (n+j)*binomial(n+j,j).
%C The former title was "Coefficients for extrapolation".
%D J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A002737/b002737.txt">Table of n, a(n) for n = 0..1000</a>
%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%F a(n) = Sum_{j=0..n} binomial(n+j,j)*(n+j). - _Zerinvary Lajos_, Aug 30 2006
%F a(n) = n*binomial(2*n+4, n+2)/4. - _Zerinvary Lajos_, Feb 28 2007
%F These 2 formulas are correct - see A331432. - _N. J. A. Sloane_, Jan 17 2020
%F a(n) = (n*(2*n + 3)*binomial(2*n + 1, n + 1))/(n + 2). - _Peter Luschny_, Jan 18 2020
%F 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
%F 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
%p t5 := n-> add(binomial(n+j,j)*(n+j),j=0..n); [seq(t5(n),n=0..40)];
%p # Alternative:
%p A002737 := n -> (n*(2*n + 3)*binomial(2*n+1, n+1))/(n + 2):
%p seq(A002737(n), n=0..25); # _Peter Luschny_, Jan 18 2020
%t Table[n(2n+3)Binomial[2n+1, n+1]/(n+2), {n, 0, 25}] (* _Vincenzo Librandi_, Jan 19 2020 *)
%o (Magma) [(n*(2*n+3)*Binomial(2*n+1, n+1))/(n+2): n in [0..30]]; // _Vincenzo Librandi_, Jan 19 2020
%o (SageMath) [n*(n+3)*catalan_number(n+2)/4 for n in (0..30)] # _G. C. Greubel_, Mar 23 2022
%Y A diagonal of A331432.
%Y Cf. A000108.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Entry revised by _N. J. A. Sloane_, Jan 18 2020