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 A110561 Denominators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217. 3
 1, 1, 1, 3, 8, 40, 180, 140, 896, 72576, 604800, 6652800, 68428800, 59304960, 726485760, 163459296000, 2324754432000, 39520825344000, 640237370572800, 579262382899200, 10532043325440000, 4644631106519040000 (list; graph; refs; listen; history; text; internal format)
 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 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: A072687 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

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