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A279361 Exponential transform of the triangular numbers.

%I A279361 #18 Feb 16 2025 08:33:37
%S A279361 1,1,4,16,80,471,3127,23059,186468,1635265,15422471,155388399,
%T A279361 1663294756,18826525771,224434810797,2808247979611,36770685485408,
%U A279361 502505495269521,7150461569849395,105723461155720879,1621191824611307436,25738508587975433251
%N A279361 Exponential transform of the triangular numbers.
%H A279361 Alois P. Heinz, <a href="/A279361/b279361.txt">Table of n, a(n) for n = 0..519</a>
%H A279361 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 A279361 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 A279361 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A279361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialTransform.html">Exponential Transform</a>
%H A279361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H A279361 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A279361 E.g.f.: exp(exp(x)*x*(x+2)/2).
%e A279361 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 A279361 a:= proc(n) option remember; `if`(n=0, 1,
%p A279361       add(binomial(n-1, j-1)*j*(j+1)/2*a(n-j), j=1..n))
%p A279361     end:
%p A279361 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 11 2016
%t A279361 Range[0, 23]! CoefficientList[Series[Exp[Exp[x] x (x + 2)/2], {x, 0, 23}], x]
%Y A279361 Cf. A000217, A033462.
%K A279361 nonn
%O A279361 0,3
%A A279361 _Ilya Gutkovskiy_, Dec 10 2016