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a(n) = Sum_{i=1..n} C(i+3,4)^3.
21

%I #22 Feb 18 2024 17:44:56

%S 1,126,3501,46376,389376,2389752,11650752,47587752,168875127,

%T 534401002,1537404003,4080706128,10109274128,23590546128,52243162128,

%U 110473767504,224205418629,438589465254,830009446129,1524339072504,2724140666880,4748425291880,8089787666880

%N a(n) = Sum_{i=1..n} C(i+3,4)^3.

%H T. D. Noe, <a href="/A086024/b086024.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 a(n) = ( C(n+4, 5)/1001 )*( 1001 +20020*C(n-1, 1) +125840*C(n-1, 2) +390390*C(n-1, 3) +695695*C(n-1, 4) +750750*C(n-1, 5) +486850*C(n-1, 6) +175175*C(n-1, 7) +26950*C(n-1, 8) ).

%F G.f.: x*(1 +112*x +1828*x^2 +8464*x^3 +13840*x^4 +8464*x^5 +1828*x^6 +112*x^7 +x^8)/(x-1)^14 . - _R. J. Mathar_, Dec 22 2013

%F -(n-1)^3*a(n) +2*(n+1)*(n^2+2*n+13)*a(n-1) -(n+3)^3*a(n-2)=0. - _R. J. Mathar_, Dec 22 2013

%F a(n) = (n/69189120)*(13824 + 960960*n^2 + 5885880*n^3 + 14370356*n^4 + 19269250*n^5 + 15996695*n^6 + 8678670*n^7 + 3138135*n^8 + 750750*n^9 + 114205*n^10 + 10010*n^11 + 385*n^12). - _G. C. Greubel_, Nov 22 2017

%t Table[(n/69189120)*(13824 + 960960*n^2 + 5885880*n^3 + 14370356*n^4 + 19269250*n^5 + 15996695*n^6 + 8678670*n^7 + 3138135*n^8 + 750750*n^9 + 114205*n^10 + 10010*n^11 + 385*n^12), {n,1,30}] (* _G. C. Greubel_, Nov 22 2017 *)

%t LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,126,3501,46376,389376,2389752,11650752,47587752,168875127,534401002,1537404003,4080706128,10109274128,23590546128},30] (* _Harvey P. Dale_, Feb 18 2024 *)

%o (PARI) for(n=1,30, print1(sum(k=1,n, binomial(k+3, 4)^3), ", ")) \\ _G. C. Greubel_, Nov 22 2017

%o (Magma) [(n/69189120)*(13824 + 960960*n^2 + 5885880*n^3 + 14370356*n^4 + 19269250*n^5 + 15996695*n^6 + 8678670*n^7 + 3138135*n^8 + 750750*n^9 + 114205*n^10 + 10010*n^11 + 385*n^12): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017

%Y Cf. A087127, A024166, A085438 - A085442, A086020 - A086030.

%K easy,nonn

%O 1,2

%A _André F. Labossière_, Jul 11 2003