A395903 - OEIS

%I #51 Jun 12 2026 18:24:54

%S 448448,4643520,18905280,56615360,142418880,318416448,652527680,

%T 1249469760,2264834880,3922790720,6537968448,10542143040,16516351040,

%U 25229131200,37681613760,55160224448,79297809600,112144029120,156245904320,214739448000,291453344448,391025687360

%N a(n) = n^8 + 60*n^7 + 1490*n^6 + 19800*n^5 + 151761*n^4 + 671580*n^3 + 1609180*n^2 + 1741200*n + 448448.

%C The product of 16 consecutive integers can always be expressed in the form x^2 - y^2.

%C This sequence gives the x-values in one such representation.

%H Paolo Xausa, <a href="/A395903/b395903.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F 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).

%F a(n) = (f(m) + g(m))/512 = (m^8 - 340*m^6 + 31926*m^4 - 862580*m^2 + 2551313)/256.

%F 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).

%F 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.

%F 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).

%e n = 1: 1*2*3*...*16 = 16! =   20922789888000 =  4643520^2 -  799680^2.

%e n = 2: 2*3*4*...*17       =  355687428096000 = 18905280^2 - 1312320^2.

%e n = 3: 3*4*5*...*18       = 3201186852864000 = 56615360^2 - 2027840^2.

%t A395903[n_] := #*(#*(#*(# + 140) + 6636) + 116080) + 448448 & [n*(n + 15)];

%t Array[A395903, 25, 0] (* _Paolo Xausa_, Jun 12 2026 *)

%o (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)

%o (Python)

%o 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

%Y Cf. A010969, A346376.

%K nonn,easy

%O 0,1

%A _Seiichi Manyama_, Jun 11 2026