OFFSET
0,1
COMMENTS
The product of 16 consecutive integers can always be expressed in the form x^2 - y^2.
This sequence gives the x-values in one such representation.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
Let m = 2*n+15, f(m) = (m^2 - 1^2) * (m^2 - 7^2) * (m^2 - 11^2) * (m^2 - 13^2) and g(m) = (m^2 - 3^2) * (m^2 - 5^2) * (m^2 - 9^2) * (m^2 - 15^2).
a(n) = (f(m) + g(m))/512 = (m^8 - 340*m^6 + 31926*m^4 - 862580*m^2 + 2551313)/256.
Product_{k=0..15} (n+k) = a(n)^2 - b(m)^2, where b(m) = (f(m) - g(m))/512 = 4 * (3*m^4 - 170*m^2 - 1513).
G.f.: 64 * (7007 + 9492*x - 105348*x^2 + 249452*x^3 - 313758*x^4 + 239610*x^5 - 112180*x^6 + 29820*x^7 - 3465*x^8) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
EXAMPLE
n = 1: 1*2*3*...*16 = 16! = 20922789888000 = 4643520^2 - 799680^2.
n = 2: 2*3*4*...*17 = 355687428096000 = 18905280^2 - 1312320^2.
n = 3: 3*4*5*...*18 = 3201186852864000 = 56615360^2 - 2027840^2.
MATHEMATICA
A395903[n_] := #*(#*(#*(# + 140) + 6636) + 116080) + 448448 & [n*(n + 15)];
Array[A395903, 25, 0] (* Paolo Xausa, Jun 12 2026 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(64*(7007+9492*x-105348*x^2+249452*x^3-313758*x^4+239610*x^5-112180*x^6+29820*x^7-3465*x^8)/(1-x)^9)
(Python)
def A395903(n): return n*(n*(n*(n*(n*(n*(n*(n+60)+1490)+19800)+151761)+671580)+1609180)+1741200)+448448 # Chai Wah Wu, Jun 11 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Jun 11 2026
STATUS
approved