A110561 Denominators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.

1, 1, 1, 3, 8, 40, 180, 140, 896, 72576, 604800, 6652800, 68428800, 59304960, 726485760, 163459296000, 2324754432000, 39520825344000, 640237370572800, 579262382899200, 10532043325440000, 4644631106519040000

Offset

0,4

Comments

The exponential generating function of the triangular numbers was given in Sloane & Plouffe as g(x) = (1 + 2x + (x^2)/2)*e^x = 1 + 3*x + 3*x^2 + (5/3)*x^3 + (5/8)*x^4 + (7/40)*x^5 + (1/896)*x^6 + (11/72576)*x^7 + ... = 1 + 3*x/1! + 6*(x^2)/2! + 10*(x^3)/3! + 15*(x^4)/4! + ...

References

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995, p. 9.

Links

Table of n, a(n) for n=0..21.

Eric Weisstein's World of Mathematics, Triangular Number.

Formula

A110560(n)/A110561(n) is the n-th coefficient of the exponential generating function of T(n), the triangular numbers A000217.

Example

a(3) = 3 because T(3+1)/3! = T(4)/3! = (4*5/2)/(1*2*3) = 10/6 = 5/3 so the fraction has denominator 3 and numerator A110560(3) = 5. Furthermore, the 3rd term of the exponential generating function of the triangular numbers is (5/3)*x^3.

Mathematica

T[n_] := n*(n + 1)/2; Table[Denominator[T[n + 1]/n! ], {n, 0, 21}]
With[{nn=30}, Denominator[Accumulate[Range[nn]]/Range[0, nn-1]!]] (* Harvey P. Dale, Aug 15 2014 *)

Crossrefs

Numerator = A110560.

Closely related to this is T(n)/n! which is A090585/A090586.

Sequence in context: A353718 A260817 A262126 * A107991 A007175 A152394

Adjacent sequences: A110558 A110559 A110560 * A110562 A110563 A110564

Keyword

easy,frac,nonn

Author

Jonathan Vos Post, Jul 27 2005

Extensions

Extended by Ray Chandler, Jul 27 2005

Status

approved