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A110560 Numerators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217. 3
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 (list; graph; refs; listen; history; text; internal format)
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

A110560(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)). [From 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: A002374 A261046 A226482 * 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

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Last modified September 24 04:27 EDT 2017. Contains 292403 sequences.