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
a(n) ~ 4^(n+1) * sqrt(n/Pi). - Amiram Eldar, Sep 25 2025
From Amiram Eldar, Jan 22 2026: (Start)
Sum_{n>=1} 1/a(n) = (13 - 2*sqrt(3)*Pi)/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(phi)/(3*sqrt(5)) - 1/9, where phi is the golden ratio (A001622). (End)
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Entry revised by N. J. A. Sloane, Jan 18 2020
STATUS
approved