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%I #27 Jun 15 2023 13:35:34
%S 0,6,126,1092,5460,19530,55986,137256,299592,597870,1111110,1948716,
%T 3257436,5229042,8108730,12204240,17895696,25646166,36012942,49659540,
%U 67368420,90054426,118778946,154764792,199411800,254313150,321272406,402321276,499738092
%N a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n.
%H Alois P. Heinz, <a href="/A228290/b228290.txt">Table of n, a(n) for n = 0..1000</a>
%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.: -6*x*(7*x^4+42*x^3+56*x^2+14*x+1)/(x-1)^7.
%F a(n) = (n+1)*(n^2+n+1)*a(n-1)/((n-1)*(n^2-3*n+3)) for n>1.
%F a(1) = 6, else a(n) = (n^7-n)/(n-1).
%F a(n) = 6*A059721(n) = n*(n+1)*(1+n+n^2)*(1-n+n^2). - _R. J. Mathar_, Aug 21 2013
%F a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7) for n>6, a(0)=0, a(1)=6, a(2)=126, a(3)=1092, a(4)=5460, a(5)=19530, a(6)=55986. - _Yosu Yurramendi_, Sep 03 2013
%p a:= n-> (1+(1+(1+(1+(1+n)*n)*n)*n)*n)*n:
%p seq(a(n), n=0..30);
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<2, 6*n,
%p (n+1)*(n^2+n+1)*a(n-1)/((n-1)*(n^2-3*n+3)))
%p end:
%p seq(a(n), n=0..30);
%p # third Maple program:
%p a:= n-> `if`(n=1, 6, (n^7-n)/(n-1)):
%p seq(a(n), n=0..30);
%o (R)
%o a <- c(0, 6, 126, 1092, 5460, 19530, 55986)
%o for(n in (length(a)+1):30) a[n] <- 7*a[n-1] -21*a[n-2] +35*a[n-3] -35*a[n-4] +21*a[n-5] -7*a[n-6] +a[n-7]
%o a
%o [_Yosu Yurramendi_, Sep 03 2013]
%o (PARI) a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n; \\ _Joerg Arndt_, Sep 03 2013
%Y Column k=6 of A228275.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Aug 19 2013
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