OFFSET
1,2
COMMENTS
a(n) is odd for n of the form 2^m or 2^m+1 (with m >= 0) (i.e., a(n) is odd if n=1,2,3,4,5,8,9,16,17,32,33,64,65,128,129,256,257...)
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..250
FORMULA
From Vaclav Kotesovec, Sep 07 2025: (Start)
Recurrence: n*(n^9 - 23*n^8 + 216*n^7 - 1086*n^6 + 3257*n^5 - 6279*n^4 + 8302*n^3 - 7260*n^2 + 2888*n + 224)*a(n) = (4*n^12 - 97*n^11 + 983*n^10 - 5494*n^9 + 18926*n^8 - 42889*n^7 + 66939*n^6 - 72992*n^5 + 52332*n^4 - 18312*n^3 - 6584*n^2 + 11264*n - 3360)*a(n-1) - (n-1)*(6*n^13 - 159*n^12 + 1813*n^11 - 11782*n^10 + 48738*n^9 - 135783*n^8 + 263033*n^7 - 361716*n^6 + 359122*n^5 - 249376*n^4 + 68336*n^3 + 98264*n^2 - 114816*n + 30240)*a(n-2) + (n-2)*(n-1)*(4*n^14 - 119*n^13 + 1569*n^12 - 12152*n^11 + 61620*n^10 - 215241*n^9 + 530125*n^8 - 931942*n^7 + 1194554*n^6 - 1164306*n^5 + 812832*n^4 - 80296*n^3 - 647608*n^2 + 617760*n - 151200)*a(n-3) - (n-3)^3*(n-2)*(n-1)*(n^13 - 28*n^12 + 353*n^11 - 2644*n^10 + 12905*n^9 - 42052*n^8 + 90903*n^7 - 131252*n^6 + 143550*n^5 - 144048*n^4 + 65872*n^3 + 128360*n^2 - 182400*n + 50400)*a(n-4) + 2*(n-4)^4*(n-3)^2*(n-2)*(n-1)*(2*n - 7)*(n^9 - 14*n^8 + 68*n^7 - 134*n^6 + 115*n^5 - 208*n^4 + 392*n^3 + 180*n^2 - 736*n + 240)*a(n-5).
a(n) ~ n^(2*n + 2/3) / (2 * sqrt(3) * exp(2*n - 3*n^(1/3))). (End)
MATHEMATICA
Table[n!*Sum[(k+n)!/(k!)^3, {k, n}]/2, {n, 25}] (* G. C. Greubel, Sep 07 2025 *)
PROG
(PARI) a(n)=if(n<0, 0, (n!/2)*sum(k=1, n, (n+k)!/(k!)^3))
(Magma)
A073014:= func< n | Factorial(n)*(&+[Factorial(n+k)/Factorial(k)^3 : k in [1..n]])/2 >;
[A073014(n): n in [1..25]]; // G. C. Greubel, Sep 07 2025
(SageMath)
def A073014(n): return sum((factorial(n)*factorial(n+k))//(factorial(k))^3 for k in range(1, n+1))//2
print([A073014(n) for n in range(1, 26)]) # G. C. Greubel, Sep 07 2025
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Aug 03 2002
STATUS
approved