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

0,2

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Triangular Number.

Formula

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]

Example

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.

Mathematica

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 *)

Crossrefs

Denominator = A110561.

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

Sequence in context: A261046 A226482 A338777 * A172170 A233808 A141424

Adjacent sequences: A110557 A110558 A110559 * A110561 A110562 A110563

Keyword

easy,frac,nonn

Author

Jonathan Vos Post, Jul 27 2005

Extensions

Extended by Ray Chandler, Jul 27 2005

Status

approved