A002739 a(n) = ((2*n-1)!/(2*n!*(n-2)!))*((n^3-3*n^2+2*n+2)/(n^2-1)). (Formerly M4732 N2024)

1, 10, 91, 651, 4026, 22737, 120835, 615043, 3031678, 14578928, 68747966, 319075550, 1461581460, 6621579135, 29718121635, 132302508195, 584868588150, 2569600678260, 11227927978410, 48822435838410, 211370463290220, 911509393468050, 3916793943349326, 16776146058210126, 71641860657928876

Offset

2,2

Comments

The former name 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 = 2..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) ~ 2^(2*n-5)*(8*n-33)*sqrt(n/Pi). - Peter Luschny, Jan 18 2020

From G. C. Greubel, Mar 23 2022: (Start)

a(n) = (1/4)*(n^3 - 3*n^2 + 2*n + 2)*A000108(n).

G.f.: (1 -9*x +21*x^2 +2*x^3)/(2*x*(1-4*x)^(5/2)) - (1 +x +x^2)/(2*x). (End)

Maple

t4 := n-> ((2*n-1)! /(2*n!*(n-2)!))*((n^3-3*n^2+2*n+2)/(n^2-1));
[seq(t4(n), n=2..40)];

Mathematica

Table[(n^3-3*n^2+2*n+2)*CatalanNumber[n]/4, {n, 2, 30}]

Prog

(Magma) [(n^3 -3*n^2 +2*n +2)*Catalan(n)/4: n in [2..30]]; // G. C. Greubel, Mar 23 2022
(Sage) [(n^3 -3*n^2 +2*n +2)*catalan_number(n)/4 for n in (2..30)] # G. C. Greubel, Mar 23 2022

Crossrefs

A diagonal of A331432.

Sequence in context: A143572 A365305 A244203 * A344389 A079928 A346230

Adjacent sequences: A002736 A002737 A002738 * A002740 A002741 A002742

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved