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A112495 Third column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.

%I A112495 #18 Mar 02 2025 23:41:31
%S A112495 3,25,130,546,2037,7071,23436,75328,237127,735813,2260518,6896046,
%T A112495 20933673,63325051,191088976,575625900,1731858075,5206059585,
%U A112495 15640198410,46966732090,140996664733,423191320215,1269993390420
%N A112495 Third column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.
%C A112495 2*a(n-4) is the number of ternary words of length n where two of the letters are used at least twice.  For example, for n=5 the 50 words that use 0 and 1 at least twice are 00011 (10 of this type), 00111 (10 of this type) and 00112 (30 of this type). - _Enrique Navarrete_, Feb 14 2025
%H A112495 G. C. Greubel, <a href="/A112495/b112495.txt">Table of n, a(n) for n = 0..1000</a>
%H A112495 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,82,-91,52,-12).
%F A112495 a(n) = 3*a(n-1)+ (n+3)*(2^(n+2)-(n+3)), n>=1, a(0)=3.
%F A112495 G.f.: (3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x)).
%F A112495 a(n) = 3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2. - _Vaclav Kotesovec_, Jul 23 2021
%F A112495 E.g.f.: (1/2)*exp(x)*(exp(x)-x-1)^2 (with offset 4). - _Enrique Navarrete_, Feb 14 2025
%t A112495 CoefficientList[Series[(3 - 5*x)/(((1 - x)^3)*((1 - 2*x)^2)*(1 - 3*x)), {x, 0, 50}], x] (* _G. C. Greubel_, Nov 13 2017 *)
%t A112495 Table[3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2, {n,0,25}] (* _Vaclav Kotesovec_, Jul 23 2021 *)
%o A112495 (PARI) x='x+O('x^50); Vec((3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x))) \\ _G. C. Greubel_, Nov 13 2017
%Y A112495 Cf. A000295 (second column).
%Y A112495 Column k=2 of A124324 (shifted).
%K A112495 nonn,easy
%O A112495 0,1
%A A112495 _Wolfdieter Lang_, Oct 14 2005