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A377662 a(n) = Sum_{k=0..n} binomial(n, k) * Sum_{j=k..n} n!/(k!*(j - k)!). Row sums of A377661.

%I A377662 #12 Mar 30 2025 20:23:40
%S A377662 1,3,14,80,534,4102,35916,354888,3915750,47754938,637840356,
%T A377662 9256590928,144977618044,2436460447020,43719637179224,834042701945520,
%U A377662 16852447379512710,359468276129261730,8070500634880125300,190211302604157871680,4695001374741310892820
%N A377662 a(n) = Sum_{k=0..n} binomial(n, k) * Sum_{j=k..n} n!/(k!*(j - k)!). Row sums of A377661.
%F A377662 a(n) = e*Sum_{k=0..n} binomial(n, k)^2 * Gamma(n - k + 1, 1).
%F A377662 a(n) = Sum_{k=0..n} binomial(n, k)^2 * KummerU(k - n, k - n, 1).
%F A377662 a(n) = Sum_{k=0..n} binomial(n, k) * A073107(n, k).
%F A377662 From _Vaclav Kotesovec_, Nov 07 2024: (Start)
%F A377662 Recurrence: n*(n^2 - 9*n + 12)*a(n) = 2*(n^4 - 7*n^3 - 5*n^2 + 29*n - 12)*a(n-1) - (n-1)*(n^4 - 2*n^3 - 47*n^2 + 104*n - 48)*a(n-2) + 2*(n-2)^2*(2*n - 3)*(n^2 - 7*n + 4)*a(n-3).
%F A377662 a(n) ~ 2^(-1/2) * exp(2*sqrt(n) - n + 1/2) * n^(n + 1/4) * (1 + 31/(48*sqrt(n))). (End)
%p A377662 A377662 := n -> add(binomial(n, k)*add(n!/(k!*(j-k)!), j = k..n), k = 0..n):
%p A377662 seq(A377662(n), n = 0..20);
%p A377662 # Or:
%p A377662 a := n -> add(binomial(n, k)^2 * KummerU(k-n, k-n, 1), k = 0..n):
%p A377662 seq(simplify(a(n)), n = 0..20);
%t A377662 a[n_] := E Sum[Gamma[n - k + 1, 1] Binomial[n, k]^2, {k, 0, n}];
%t A377662 Table[a[n], {n, 0, 20}]
%Y A377662 Cf. A377661, A073107.
%K A377662 nonn
%O A377662 0,2
%A A377662 _Peter Luschny_, Nov 07 2024