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A359739 a(n) = Sum_{j=0..n, j even} binomial(n, j) * oddfactorial(j/2) * n^j, where oddfactorial(n) = (2*n)! / (2^n*n!).

%I A359739 #9 Jan 18 2023 09:34:20
%S A359739 1,1,5,28,865,9626,758701,12606280,1872570113,41351249980,
%T A359739 9925656304501,273345587759696,96567039881462305,3185756105692821688,
%U A359739 1555524449985942662045,59790093545794928817376,38565845285812075675023361,1692346747225524397926264080,1393672439437011815394433559653
%N A359739 a(n) = Sum_{j=0..n, j even} binomial(n, j) * oddfactorial(j/2) * n^j, where oddfactorial(n) = (2*n)! / (2^n*n!).
%F A359739 Let K(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)) denote the Kummer polynomials, defined in A359760.
%F A359739 a(n) = K(n, n) for n >= 1.
%p A359739 A359739 := n -> ifelse(n=0, 1, KummerU(-n/2, 1/2, -1/(2*n^2))*(-1/(2*n^2))^(-n/2)): seq(simplify(A359739(n)), n = 0..18);
%o A359739 (Python)
%o A359739 from math import factorial, comb
%o A359739 def oddfactorial(n: int) -> int: return factorial(2 * n) // (2**n * factorial(n))
%o A359739 def a(n: int) -> int:
%o A359739     return sum(comb(n, j) * oddfactorial(j//2) * n**j for j in range(0, n+1, 2))
%o A359739 print([a(n) for n in range(19)])
%Y A359739 Cf. A359760.
%K A359739 nonn
%O A359739 0,3
%A A359739 _Peter Luschny_, Jan 12 2023