A279361 Exponential transform of the triangular numbers.

1, 1, 4, 16, 80, 471, 3127, 23059, 186468, 1635265, 15422471, 155388399, 1663294756, 18826525771, 224434810797, 2808247979611, 36770685485408, 502505495269521, 7150461569849395, 105723461155720879, 1621191824611307436, 25738508587975433251

Offset: 0

Author

Ilya Gutkovskiy, Dec 10 2016

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x/1! + 4*x^2/2! + 16*x^3/3! + 80*x^4/4! + 471*x^5/5! + 3127*x^6/6! + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..519
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Exponential Transform
• Eric Weisstein's World of Mathematics, Triangular Number
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000217, A033462.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j*(j+1)/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 11 2016

Mathematica

Range[0, 23]! CoefficientList[Series[Exp[Exp[x] x (x + 2)/2], {x, 0, 23}], x]

Formula

E.g.f.: exp(exp(x)*x*(x+2)/2).