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A343276 a(n) = n! * [x^n] -x*(x + 1)*exp(x)/(x - 1)^3.

%I A343276 #17 Jun 15 2025 12:40:32
%S A343276 0,1,10,81,652,5545,50886,506905,5480056,64116657,808856290,
%T A343276 10959016321,158851484100,2454385635481,40285778016862,
%U A343276 700261611998985,12853532939027056,248482678808005345,5047002269952482106,107466341437781300017,2394019421567804960380
%N A343276 a(n) = n! * [x^n] -x*(x + 1)*exp(x)/(x - 1)^3.
%F A343276 a(n) = Sum_{k=0..n} rf(n - k + 1, k)*k^2, where rf is the rising factorial.
%F A343276 a(n) = (2 + n*(n + 2))*a(n - 1)/(n - 1) - (n + 1)*a(n - 2) for n >= 3.
%F A343276 A002775(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(k).
%F A343276 a(n) = Sum_{k=1..n} k^2*k!*binomial(n,k). - _Ridouane Oudra_, Jun 15 2025
%p A343276 egf := -x*(x + 1)*exp(x)/(x - 1)^3: ser := series(egf, x, 32):
%p A343276 seq(n!*coeff(ser, x, n), n = 0..20);
%t A343276 a[n_] := Sum[Pochhammer[n - k + 1, k]*k^2, {k, 0, n}];
%t A343276 Table[a[n], {n, 0, 20}]
%o A343276 (SageMath)
%o A343276 def a(n): return sum(rising_factorial(n - k + 1, k)*k^2 for k in (0..n))
%o A343276 print([a(n) for n in (0..20)])
%o A343276 (Python)
%o A343276 def a():
%o A343276     a, b, n = 0, 1, 2
%o A343276     yield 0
%o A343276     while True:
%o A343276         yield b
%o A343276         a, b = b, -(n + 1)*a + ((2 + n*(n + 2))*b)//(n - 1)
%o A343276         n += 1
%o A343276 A343276 = a(); print([next(A343276) for _ in range(21)])
%Y A343276 Cf. A002775, A093964.
%K A343276 nonn
%O A343276 0,3
%A A343276 _Peter Luschny_, Apr 20 2021