cp's OEIS Frontend

A281231 Exponential transform of the tetrahedral numbers (A000292).

%I A281231 #13 Feb 16 2025 08:33:39
%S A281231 1,1,5,23,133,916,7107,61286,580505,5968400,66032901,780962524,
%T A281231 9817927385,130572957724,1829676460991,26919714974436,414591408939313,
%U A281231 6665930432840304,111624874150941193,1942675652654112012,35071252458352443001,655641049733709757516
%N A281231 Exponential transform of the tetrahedral numbers (A000292).
%H A281231 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A281231 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 A281231 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A281231 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialTransform.html">Exponential Transform</a>
%H A281231 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>
%H A281231 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F A281231 E.g.f.: exp(exp(x)*x*(1+x+x^2/6)).
%e A281231 E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 23*x^3/3! + 133*x^4/4! + 916*x^5/5! + 7107*x^6/6! + ...
%p A281231 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p A281231       *binomial(n-1, j-1)*j*(j+1)*(j+2)/6, j=1..n))
%p A281231     end:
%p A281231 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 18 2017
%t A281231 Range[0, 21]! CoefficientList[Series[Exp[Exp[x] x (1 + x + x^2/6)], {x, 0, 21}], x]
%Y A281231 Cf. A000292, A279361, A279358.
%K A281231 nonn
%O A281231 0,3
%A A281231 _Ilya Gutkovskiy_, Jan 18 2017