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a(n) = Sum_{i=1..n} binomial(i+1,2)^4.
19

%I #23 Sep 08 2022 08:45:11

%S 1,82,1378,11378,62003,256484,871140,2550756,6651381,15802006,

%T 34776742,71791798,140366759,261917384,469277384,811379400,1359360681,

%U 2214396762,3517606762,5462416762,8309813083,12406965164,18209748140,26309748140,37466388765,52644875166

%N a(n) = Sum_{i=1..n} binomial(i+1,2)^4.

%H G. C. Greubel, <a href="/A085439/b085439.txt">Table of n, a(n) for n = 1..5000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F a(n) = (2520*n^9 +22680*n^8 +79920*n^7 +136080*n^6 +107352*n^5 +22680*n^4 -10080*n^3 +1728*n)/9!.

%F G.f.: x*(x^6+72*x^5+603*x^4+1168*x^3+603*x^2+72*x+1) / (x-1)^10. - _Colin Barker_, May 02 2014

%e a(15) = (2520*(15^9) +22680*(15^8) +79920*(15^7) +136080*(15^6) +107352*(15^5) +22680*(15^4) -10080*(15^3) +1728*15)/9! = 469277384.

%t Table[(2520*(n^9) + 22680*(n^8) + 79920*(n^7) + 136080*(n^6) + 107352*(n^5) + 22680*(n^4) - 10080*(n^3) + 1728*n)/9!, {n, 1, 50}] (* _G. C. Greubel_, Nov 22 2017 *)

%o (PARI) Vec(x*(x^6+72*x^5+603*x^4+1168*x^3+603*x^2+72*x+1)/(x-1)^10 + O(x^100)) \\ _Colin Barker_, May 02 2014

%o (PARI) a(n) = sum(i=1, n, binomial(i+1, 2)^4); \\ _Michel Marcus_, Nov 22 2017

%o (Magma) [(2520*n^9 +22680*n^8 +79920*n^7 +136080*n^6 +107352*n^5 +22680*n^4 -10080*n^3 +1728*n)/Factorial(9): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017

%Y Column k=4 of A334781.

%Y Cf. A000292, A087127, A024166, A024166, A085438, A085440, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

%K easy,nonn

%O 1,2

%A _André F. Labossière_, Jul 03 2003

%E More terms from _Colin Barker_, May 02 2014

%E Typo in example fixed by _Colin Barker_, May 02 2014