A110561 -id:A110561 - cp's OEIS frontend

A110560 Numerators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.

1, 3, 3, 5, 5, 7, 7, 1, 1, 11, 11, 13, 13, 1, 1, 17, 17, 19, 19, 1, 1, 23, 23, 1, 1, 1, 1, 29, 29, 31, 31, 1, 1, 1, 1, 37, 37, 1, 1, 41, 41, 43, 43, 1, 1, 47, 47, 1, 1, 1, 1, 53, 53, 1, 1, 1, 1, 59, 59, 61, 61, 1, 1, 1, 1, 67, 67, 1, 1, 71, 71, 73, 73, 1, 1, 1, 1, 79, 79, 1, 1, 83, 83

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) = 5 because T(3+1)/3! = T(4)/3! = (4*5/2)/(1*2*3) = 10/6 = 5/3 so the fraction has numerator 5 and denominator A110561(3) = 3. 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.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Triangular Number.

Denominator = A110561.

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

T[n_] := n*(n + 1)/2; Table[Numerator[T[n + 1]/n! ], {n, 0, 82}]
Join[{1},Numerator[With[{nn=90},Rest[Accumulate[Range[nn+1]]]/ Range[ nn]!]]] (* Harvey P. Dale, Feb 17 2016 *)

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

a(n) = Denominator((n+2)!*HarmonicNumber(n+2)/binomial(n+2,2)). [Gary Detlefs, Dec 03 2011]

Extended by Ray Chandler, Jul 27 2005

cp's OEIS Frontend

A110560 Numerators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.

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