A002737 a(n) = Sum_{j=0..n} (n+j)*binomial(n+j,j). (Formerly M3975 N1644)

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

0,2

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