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a(n) = RF(n+1,3)*C(n+2,n-1), where RF(a,n) is the rising factorial.
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%I #25 Aug 05 2024 12:10:50

%S 0,24,240,1200,4200,11760,28224,60480,118800,217800,377520,624624,

%T 993720,1528800,2284800,3329280,4744224,6627960,9097200,12289200,

%U 16364040,21507024,27931200,35880000,45630000,57493800,71823024,89011440,109498200,133771200,162370560

%N a(n) = RF(n+1,3)*C(n+2,n-1), where RF(a,n) is the rising factorial.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F G.f.: -24/(x-1)^4 - 144/(x-1)^5 - 240/(x-1)^6 - 120/(x-1)^7. See the comment in A253284 for the general case.

%F a(n) = n*((n+1)*(n+2))^2*(n+3)/6.

%F a(n) = (N^3 + 4*N^2 + 4*N)/6 = N*(N + 2)^2/6 with N = n^2 + 3*n.

%F From _Bruno Berselli_, Mar 06 2018: (Start)

%F a(n) = 24*A006542(n+3) for n>0.

%F a(n) = Sum_{i=0..n} i*(i+1)^3*(i+2). Therefore, the first differences are in A133754. (End)

%p seq(n*((n+1)*(n+2))^2*(n+3)/6,n=0..19);

%t Table[n ((n + 1) (n + 2))^2 (n + 3)/6, {n, 0, 40}] (* _Bruno Berselli_, Mar 06 2018 *)

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,24,240,1200,4200,11760,28224},40] (* _Harvey P. Dale_, Aug 05 2024 *)

%o (Sage) [n*((n+1)*(n+2))^2*(n+3)/6 for n in (0..40)] # _Bruno Berselli_, Mar 06 2018

%o (GAP) List([0..40], n -> n*((n+1)*(n+2))^2*(n+3)/6); # _Bruno Berselli_, Mar 06 2018

%o (Magma) [n*((n+1)*(n+2))^2*(n+3)/6: n in [0..40]]; // _Bruno Berselli_, Mar 06 2018

%o (Python) [n*((n+1)*(n+2))**2*(n+3)/6 for n in range(40)] # _Bruno Berselli_, Mar 06 2018

%Y Cf. A002378, A006542, A083374, A133754, A253284.

%K nonn,easy

%O 0,2

%A _Peter Luschny_, Mar 23 2015