%I #49 Sep 08 2022 08:45:11

%S 1,730,47386,1047386,12438011,98204132,580094436,2756876772,

%T 11060642397,38741283022,121395233038,346594833742,914464085783,

%U 2254559726408,5240543726408,11568062614344,24395756421273,49397866465794,96443747465794,182209868465794

%N a(n) = Sum_{i=1..n} binomial(i+1,2)^6.

%H T. D. Noe, <a href="/A085441/b085441.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14, -91, 364, -1001, 2002, -3003, 3432, -3003, 2002, -1001, 364, -91, 14, -1).

%F G.f.: x*(x^10 +716*x^9 +37257*x^8 +450048*x^7 +1822014*x^6 +2864328*x^5 +1822014*x^4 +450048*x^3 +37257*x^2 +716*x +1) / (x -1)^14. - _Colin Barker_, May 02 2014

%F a(n) = (n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12). - _G. C. Greubel_, Nov 22 2017

%e a(5) = C(7,3)*[191*106 + 450*(18*C(14,10) + 3851*C(13,10) + 61839*C(12,10) + 225352*C(11,10) + 225352*C(10,10))]/10010 = 12438011.

%p f:= sum(binomial(1+i,2)^6,i=1..n):

%p seq(f, n=1..30); # _Robert Israel_, Nov 22 2017

%t Table[Sum[Binomial[i+1,2]^6,{i,n}],{n,20}] (* or *) LinearRecurrence[ {14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,730,47386,1047386,12438011, 98204132,580094436, 2756876772,11060642397, 38741283022,121395233038, 346594833742, 914464085783, 2254559726408},20] (* _Harvey P. Dale_, Jun 05 2017 *)

%o (PARI) for(n=1,30, print1(sum(k=1,n, binomial(k+1,2)^6), ", ")) \\ _G. C. Greubel_, Nov 22 2017

%o (Magma) [(n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017

%Y Column k=6 of A334781.

%Y Cf. A000292, A087127, A024166, A024166, A085438, A085439, A085440, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030, A234253.

%K easy,nonn

%O 1,2

%A _André F. Labossière_, Jul 07 2003