A110560 -id:A110560 - cp's OEIS frontend

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

Jonathan Vos Post, Jul 27 2005

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! + ...

			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.

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

Eric Weisstein's World of Mathematics, Triangular Number.

Numerator = A110560.

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

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 *)

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

Extended by Ray Chandler, Jul 27 2005

cp's OEIS Frontend

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

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