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A350267 a(n) = n*hypergeom([1, 1 - n, -n], [2], 1).

%I A350267 #30 Jul 01 2025 07:31:13
%S A350267 0,1,4,18,100,675,5376,49294,510728,5894109,74915740,1039180186,
%T A350267 15613569324,252501251743,4371586879128,80652138666870,
%U A350267 1579212732426256,32701859350855769,713914404925713588,16384896394304282722,394340620941231415540,9929838681717090607611
%N A350267 a(n) = n*hypergeom([1, 1 - n, -n], [2], 1).
%F A350267 a(n) = n*A247499(n - 1) for n >= 1.
%F A350267 a(n) = Sum_{k=1..n} binomial(n, k)^2 * k! / (n - k + 1).
%F A350267 E.g.f.: (exp(x/(1-x)) - exp(x))/x. - _Vladimir Kruchinin_, _Seiichi Manyama_, Jul 01 2025
%p A350267 A350267 := n -> n*hypergeom([1, 1 - n, -n], [2], 1): seq(simplify(A350267(n)), n = 0..21);
%p A350267 # Or:
%p A350267 egf := (exp(x/(1-x)) - exp(x))/x: ser := series(egf, x, 23):
%p A350267 seq(n!*coeff(ser, x, n), n = 0..21);  # _Peter Luschny_, Jul 01 2025
%t A350267 a[n_] := Sum[Binomial[n, k]^2 * k!/(n - k + 1), {k, 1, n}]; Array[a, 22, 0] (* _Amiram Eldar_, Jan 09 2022 *)
%o A350267 (PARI) a(n) = sum(k=1, n, binomial(n, k)^2 * k! / (n - k + 1)); \\ _Michel Marcus_, Jan 09 2022
%Y A350267 Row sums of A350266.
%Y A350267 Cf. A247499.
%K A350267 nonn
%O A350267 0,3
%A A350267 _Peter Luschny_, Jan 09 2022
%E A350267 Definition changed to a(0) = 0 by _Peter Luschny_, Jul 01 2025