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%I #30 Dec 31 2023 10:05:54
%S 1,5,21,94,485,2931,20545,164396,1479609,14796145,162757661,
%T 1953092010,25390196221,355462747199,5331941208105,85311059329816,
%U 1450288008607025,26105184154926621,495998498943605989,9919969978872119990
%N a(1) = 1, a(n) = Sum_{k=1..n} a(n-1) + k.
%C Eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, the triangular series (1, 3, 6, ...) as the left border; and the rest zeros. - _Gary W. Adamson_, Aug 01 2016
%H Harvey P. Dale, <a href="/A103519/b103519.txt">Table of n, a(n) for n = 1..449</a>
%F a(n+1) = k*(k+1)/2 - a(n)*(a(n)+1)/2, where k = a(n) + n + 1.
%F a(n) = Sum_{i=0..n} (n!/(n-i)!) * (n-i)(n-i+1)/2 = Sum_{i=0..n} (n!/(n-i)!) * A000217(n-i). For n > 2, a(n) = (3*n*(n-1)/2)*floor((n-2)!*e) + n, where e=exp(1). - _Max Alekseyev_, Feb 14 2005
%F a(n) = n*a(n-1) + n*(n+1)/2. - _Emeric Deutsch_, Mar 16 2008
%F a(n) ~ 3*sqrt(Pi/2)*exp(1)*n^n*sqrt(n)/exp(n). - _Ilya Gutkovskiy_, Aug 02 2016
%F E.g.f.: x * (1+x/2) * exp(x) / (1-x). - _Seiichi Manyama_, Dec 31 2023
%e a(2) = 2 + 3 = 5, a(3) = 6 + 7 + 8 = 21, a(4) = 22 + 23 + 24 + 25.
%p a[1]:=1: for n from 2 to 20 do a[n]:=n*a[n-1]+(1/2)*n*(n+1) end do: seq(a[n], n=1..20); # _Emeric Deutsch_, Mar 16 2008
%t RecurrenceTable[{a[1]==1,a[n]==n*a[n-1]+(n(n+1))/2},a,{n,20}] (* _Harvey P. Dale_, Nov 05 2013 *)
%o (PARI) { t(n) = n*(n+1)/2 } { a(n) = sum(i=0,n,n!/(n-i)!*t(n-i)) } { a2(n) = 3*t(n-1)*floor((n-2)!*exp(1))+n } \\ _Max Alekseyev_, Feb 14 2005
%Y Cf. A007526, A103520.
%K easy,nonn
%O 1,2
%A _Amarnath Murthy_, Feb 10 2005
%E More terms from _Max Alekseyev_, Feb 14 2005
%E Name clarified by _Seiichi Manyama_, Dec 31 2023
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