OFFSET
1,2
COMMENTS
a(n) is odd for n = 1,3 and if n is of the form 2^m or 2^m+1 (with m>= 2) (i.e., a(n) is odd if n=1,3,4,5,8,9,16,17,32,33,64,65,128,129,256,257...).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..440
FORMULA
From Vaclav Kotesovec, Sep 07 2025: (Start)
Recurrence: n*(n^4 - 18*n^3 + 103*n^2 - 226*n + 164)*a(n) = (3*n^6 - 51*n^5 + 261*n^4 - 421*n^3 - 112*n^2 + 632*n - 240)*a(n-1) - (3*n^7 - 48*n^6 + 206*n^5 - 6*n^4 - 1773*n^3 + 4046*n^2 - 3412*n + 960)*a(n-2) + (n-2)^2*(n^6 - 7*n^5 - 77*n^4 + 751*n^3 - 2100*n^2 + 2224*n - 720)*a(n-3) - 2*(n-3)^2*(n-2)*(2*n - 5)*(n^4 - 14*n^3 + 55*n^2 - 70*n + 24)*a(n-4).
a(n) ~ n^(n + 1/4) / (2^(3/2) * exp(n - 2*sqrt(n) - 1/2)).
(End)
MATHEMATICA
Table[(Sum[(n+k)!/(k!)^2, {k, n}])/2, {n, 20}] (* Harvey P. Dale, Jun 14 2022 *)
(* or *)
RecurrenceTable[{2 (-3 + n)^2 (-2 + n) (-5 + 2 n) (24 - 70 n + 55 n^2 - 14 n^3 + n^4) a[-4 + n] - (-2 + n)^2 (-720 + 2224 n - 2100 n^2 + 751 n^3 - 77 n^4 - 7 n^5 + n^6) a[-3 + n] + (960 - 3412 n + 4046 n^2 - 1773 n^3 - 6 n^4 + 206 n^5 - 48 n^6 + 3 n^7) a[-2 + n] + (240 - 632 n + 112 n^2 + 421 n^3 - 261 n^4 + 51 n^5 - 3 n^6) a[-1 + n] + n (164 - 226 n + 103 n^2 - 18 n^3 + n^4) a[n] == 0, a[1] == 1, a[2] == 6, a[3] == 37, a[4] == 255}, a, {n, 1, 25}] (* Vaclav Kotesovec, Sep 07 2025 *)
PROG
(PARI) a(n)=if(n<0, 0, sum(k=1, n, (n+k)!/(k!)^2)/2)
(Magma)
A073013:= func< n | (&+[Factorial(n+k)/Factorial(k)^2: k in [1..n]])/2 >;
[A073013(n): n in [1..25]]; // G. C. Greubel, Sep 07 2025
(SageMath)
def A073013(n): return sum(factorial(n+k)//(factorial(k))^2 for k in range(1, n+1))//2
print([A073013(n) for n in range(1, 26)]) # G. C. Greubel, Sep 07 2025
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Aug 03 2002
EXTENSIONS
More terms from Wesley Ivan Hurt, Dec 26 2023
STATUS
approved