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%I #58 Jul 03 2021 12:16:30
%S 1,2,7,76,1301,26406,619207,16652168,508596489,17457431050,
%T 666726681611,28076838451212,1293333060096013,64713740778086414,
%U 3495868307630899215,202800355058036736016,12574907509808996352017,829987773918052958208018,58100729276791270637568019
%N Row sums of A174790.
%C Except for n = 2, 3, 4 and 9, the A055642(n) least significant digits of a(n) give the decimal expansion of n + 1. - _Stefano Spezia_, Jul 02 2021
%H Stefano Spezia, <a href="/A320329/b320329.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = Sum_{m=0..n} 1 + (binomial(n, m) - 1)*(n!)^2/(m!*(n - m)!).
%F a(0) = 1, a(n) = 1 + n - 2^n*n! + 2*(2*n - 1)!/(n - 1)! for n > 0.
%F From _Stefano Spezia_, Jul 02 2021: (Start)
%F E.g.f.: 1/sqrt(1 - 4*x) + exp(x)*(1 + x) + 1/(2*x - 1).
%F a(n) ~ sqrt(2)*4^n*exp(-n)*n^n. (End)
%p a := n -> add(1+(binomial(n, m)-1)*(n!)^2/(m!*(n-m)!), m = 0 .. n): seq(a(n), n = 0 .. 20);
%t T[n_, m_] = 1+((Binomial[n, m]-1)(n!)^2)/(m!(n-m)!); Table[Sum[T[n,m], {m,0,n}], {n,0,20}] (* or *)
%t a[n_]:=1+n-2^n n!+2(2n-1)!/(n-1)!; Join[{1},Array[a,20]]
%o (GAP) List([0..20], n->Sum([0..n], m->1+((Binomial(n, m)-1)*(Factorial(n)^2)/(Factorial(m)*Factorial(n-m)))));
%o (PARI) a(n) = sum(m=0, n, 1+(binomial(n, m)-1)*(n!)^2/(m!*(n-m)!));
%Y Cf. A000079, A000142, A000302, A174790.
%K nonn
%O 0,2
%A _Stefano Spezia_, Dec 18 2018
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