A279361 - OEIS

%I #18 Feb 16 2025 08:33:37

%S 1,1,4,16,80,471,3127,23059,186468,1635265,15422471,155388399,

%T 1663294756,18826525771,224434810797,2808247979611,36770685485408,

%U 502505495269521,7150461569849395,105723461155720879,1621191824611307436,25738508587975433251

%N Exponential transform of the triangular numbers.

%H Alois P. Heinz, <a href="/A279361/b279361.txt">Table of n, a(n) for n = 0..519</a>

%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialTransform.html">Exponential Transform</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

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

%e 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! + ...

%p a:= proc(n) option remember; `if`(n=0, 1,

%p       add(binomial(n-1, j-1)*j*(j+1)/2*a(n-j), j=1..n))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 11 2016

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

%Y Cf. A000217, A033462.

%K nonn,changed

%O 0,3

%A _Ilya Gutkovskiy_, Dec 10 2016