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A245176 a(n) = 2*a(n-1)+(n-2)*a(n-2)-(n-1)*a(n-3) with initial terms (1,2,4).

%I A245176 #46 Jun 30 2025 19:18:14
%S A245176 1,2,4,8,18,44,120,352,1116,3736,13232,48928,189080,757584,3148064,
%T A245176 13497600,59704336,271503648,1268817472,6078518912,29837183008,
%U A245176 149789875904,768674514816,4026518397440,21518708975040,117199152735616,650184360936192,3670861106911744
%N A245176 a(n) = 2*a(n-1)+(n-2)*a(n-2)-(n-1)*a(n-3) with initial terms (1,2,4).
%H A245176 Alois P. Heinz, <a href="/A245176/b245176.txt">Table of n, a(n) for n = 0..800</a>
%H A245176 Tewodros Amdeberhan and Victor H. Moll, <a href="https://arxiv.org/abs/1406.2356">Involutions and their progenies</a>, arXiv:1406.2356 [math.CO], 2014.
%H A245176 Carlos M. da Fonseca and Anthony G. Shannon, <a href="https://doi.org/10.47974/JIM-2148">Telephone numbers extensions</a>, J. Interdisc. Math. (2025) Vol. 28, No. 4, 1573-1580. See p. 1577.
%F A245176 a(n) = Sum_{k=0..n} A000085(k). - _Emanuele Munarini_, Aug 31 2017
%F A245176 a(n) ~ exp(sqrt(n) - n/2 - 1/4) * n^(n/2) / sqrt(2). - _Vaclav Kotesovec_, Jun 12 2018
%p A245176 a:= proc(n) option remember; 2*a(n-1)+(n-2)*a(n-2)-(n-1)*a(n-3) end:
%p A245176 a(0), a(1), a(2):= 1, 2, 4:
%p A245176 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 19 2014
%t A245176 a[n_] := Sum[StirlingS1[j, k]*2^k*BellB[k, 1/2], {j, 0, n}, {k, 0, j}];
%t A245176 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 12 2018, after _Emanuele Munarini_ *)
%o A245176 (Magma) I:=[1,2,4]; [n le 3 select I[n] else 2*Self(n-1)-2*Self(n-2)+Self(n-3)+(n-1)*(Self(n-2)-Self(n-3)): n in [1..30]]; // _Vincenzo Librandi_, Jul 22 2014 // Uses incorrect offset
%Y A245176 Partial sums of A000085.
%K A245176 nonn
%O A245176 0,2
%A A245176 _N. J. A. Sloane_, Jul 19 2014
%E A245176 More terms from _Vincenzo Librandi_, Jul 22 2014