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a(n) = n*(7*n^2 + 15*n + 8)/6.
5

%I #27 Sep 08 2022 08:46:08

%S 0,5,22,58,120,215,350,532,768,1065,1430,1870,2392,3003,3710,4520,

%T 5440,6477,7638,8930,10360,11935,13662,15548,17600,19825,22230,24822,

%U 27608,30595,33790,37200,40832,44693,48790,53130,57720,62567,67678,73060,78720,84665

%N a(n) = n*(7*n^2 + 15*n + 8)/6.

%C Row sums of the triangle in A245300.

%H G. C. Greubel, <a href="/A245301/b245301.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = n*(n+1)*(7*n+8)/6 = A002378(n)*A016993(n+1)/6.

%F a(n) = Sum_{j=0..n} A000217(2n-j)+j. - _Manfred Arens_, Dec 26 2015

%F G.f.: x*(5 + 2*x)/(1-x)^4. - _Vincenzo Librandi_, Feb 01 2016

%F E.g.f.: x*(30 + 36*x + 7*x^2)*exp(x)/6. - _G. C. Greubel_, Mar 31 2021

%p A245301:= n-> n*(n+1)*(7*n+8)/6; seq(A245301(n), n=0..50); # _G. C. Greubel_, Mar 31 2021

%t Table[n (7 n^2 + 15 n + 8)/6, {n, 0, 50}] (* _Vincenzo Librandi_, Feb 01 2016 *)

%t LinearRecurrence[{4,-6,4,-1},{0,5,22,58},50] (* _Harvey P. Dale_, Sep 21 2019 *)

%o (Haskell)

%o a245301 n = n * (n * (7 * n + 15) + 8) `div` 6

%o (PARI) a(n)=n*(7*n^2+15*n+8)/6 \\ _Charles R Greathouse IV_, Feb 01 2016

%o (Magma) [n*(7*n^2+15*n+8)/6: n in [0..60]]; // _Vincenzo Librandi_, Feb 01 2016

%o (Sage) [n*(n+1)*(7*n+8)/6 for n in (0..50)] # _G. C. Greubel_, Mar 31 2021

%Y Cf. A002378, A016993, A245300, A254407.

%K nonn,easy

%O 0,2

%A _Reinhard Zumkeller_, Jul 17 2014