A002737 a(n) = Sum_{j=0..n} (n+j)*binomial(n+j,j).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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)

Crossrefs

A diagonal of A331432.

Cf. A000108.

Programs

Magma

[(n*(2*n+3)*Binomial(2*n+1, n+1))/(n+2): n in [0..30]]; // Vincenzo Librandi, Jan 19 2020

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 *)

SageMath

[n*(n+3)*catalan_number(n+2)/4 for n in (0..30)] # G. C. Greubel, Mar 23 2022

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

Extensions

Entry revised by N. J. A. Sloane, Jan 18 2020