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

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

Author

N. J. A. Sloane

Keywords

• nonn

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)

Crossrefs

A diagonal of A331432.

Programs

Magma

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

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}]

Sage

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

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)

Extensions

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