A253285 a(n) = RF(n+1,3)*C(n+2,n-1), where RF(a,n) is the rising factorial.
0, 24, 240, 1200, 4200, 11760, 28224, 60480, 118800, 217800, 377520, 624624, 993720, 1528800, 2284800, 3329280, 4744224, 6627960, 9097200, 12289200, 16364040, 21507024, 27931200, 35880000, 45630000, 57493800, 71823024, 89011440, 109498200, 133771200, 162370560
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Programs
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GAP
List([0..40], n -> n*((n+1)*(n+2))^2*(n+3)/6); # Bruno Berselli, Mar 06 2018
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Magma
[n*((n+1)*(n+2))^2*(n+3)/6: n in [0..40]]; // Bruno Berselli, Mar 06 2018
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Maple
seq(n*((n+1)*(n+2))^2*(n+3)/6,n=0..19);
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Mathematica
Table[n ((n + 1) (n + 2))^2 (n + 3)/6, {n, 0, 40}] (* Bruno Berselli, Mar 06 2018 *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,24,240,1200,4200,11760,28224},40] (* Harvey P. Dale, Aug 05 2024 *) -
Python
[n*((n+1)*(n+2))**2*(n+3)/6 for n in range(40)] # Bruno Berselli, Mar 06 2018
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Sage
[n*((n+1)*(n+2))^2*(n+3)/6 for n in (0..40)] # Bruno Berselli, Mar 06 2018
Formula
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.
a(n) = n*((n+1)*(n+2))^2*(n+3)/6.
a(n) = (N^3 + 4*N^2 + 4*N)/6 = N*(N + 2)^2/6 with N = n^2 + 3*n.
From Bruno Berselli, Mar 06 2018: (Start)
a(n) = 24*A006542(n+3) for n>0.
a(n) = Sum_{i=0..n} i*(i+1)^3*(i+2). Therefore, the first differences are in A133754. (End)