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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!).
2
1, 1, 5, 28, 865, 9626, 758701, 12606280, 1872570113, 41351249980, 9925656304501, 273345587759696, 96567039881462305, 3185756105692821688, 1555524449985942662045, 59790093545794928817376, 38565845285812075675023361, 1692346747225524397926264080, 1393672439437011815394433559653
OFFSET
0,3
FORMULA
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.
a(n) = K(n, n) for n >= 1.
MAPLE
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);
PROG
(Python)
from math import factorial, comb
def oddfactorial(n: int) -> int: return factorial(2 * n) // (2**n * factorial(n))
def a(n: int) -> int:
return sum(comb(n, j) * oddfactorial(j//2) * n**j for j in range(0, n+1, 2))
print([a(n) for n in range(19)])
CROSSREFS
Cf. A359760.
Sequence in context: A249784 A359649 A346312 * A344464 A348393 A057642
KEYWORD
nonn
AUTHOR
Peter Luschny, Jan 12 2023
STATUS
approved