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A300455 Logarithmic transform of the triangular numbers A000217.

%I A300455 #11 Feb 16 2025 08:33:53
%S A300455 0,1,2,-1,-11,19,201,-764,-7426,52137,448435,-5377604,-38712486,
%T A300455 777663613,4258812299,-149524753650,-505685566184,36733876797025,
%U A300455 30910872539763,-11174584391207360,25170998506744790,4101787001153848461,-24862093152821214653,-1776483826032814964966
%N A300455 Logarithmic transform of the triangular numbers A000217.
%H A300455 Alois P. Heinz, <a href="/A300455/b300455.txt">Table of n, a(n) for n = 0..450</a>
%H A300455 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 A300455 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 A300455 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A300455 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicTransform.html">Logarithmic Transform</a>
%H A300455 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H A300455 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A300455 E.g.f.: log(1 + exp(x)*x*(x + 2)/2).
%e A300455 E.g.f.: A(x) = x/1! + 2*x^2/2! - x^3/3! - 11*x^4/4! + 19*x^5/5! + 201*x^6/6! - 764*x^7/7! - 7426*x^8/8! + ...
%p A300455 a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
%p A300455       binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->i*(i+1)/2)
%p A300455     end:
%p A300455 seq(a(n), n=0..25);  # _Alois P. Heinz_, Mar 06 2018
%t A300455 nmax = 23; CoefficientList[Series[Log[1 + Exp[x] x (x + 2)/2], {x, 0, nmax}], x] Range[0, nmax]!
%Y A300455 Cf. A000217, A009306, A033464, A279361, A300452.
%K A300455 sign
%O A300455 0,3
%A A300455 _Ilya Gutkovskiy_, Mar 06 2018