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%I #36 Sep 08 2022 08:45:53
%S 6,42,156,420,930,1806,3192,5256,8190,12210,17556,24492,33306,44310,
%T 57840,74256,93942,117306,144780,176820,213906,256542,305256,360600,
%U 423150,493506,572292,660156,757770,865830,985056,1116192,1260006
%N a(n) = n^4 + 6n^3 + 14n^2 + 15n + 6.
%C Essentially partial sums of A061804.
%C Agrees with the known terms listed in A082986. Are the sequences identical?
%C Partial sums of A061804 (see above comment) = 1*n^4 + 2*n^3 + 2*n^2 + 1*n^1. To obtain this sequence, all elements of which are pronic numbers of pronic number index number (e.g., a(8) = 8190 is the 90th pronic number and 90 is the 9th pronic number; 9 = 8 + 1), then switch n to (n+1). - _Raphie Frank_, Oct 17 2012
%H Vincenzo Librandi, <a href="/A176780/b176780.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: 6*(1+x)^2/(1-x)^5.
%F a(n) = 6*A006325(n+2).
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24; a(0)=6, a(1)=42, a(2)=156, a(3)=420.
%F a(n) = a(-n-3). - _Bruno Berselli_, Sep 05 2011
%t Table[n^4+6n^3+14n^2+15n+6,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{6,42,156,420,930},40] (* _Harvey P. Dale_, Mar 28 2012 *)
%o (Magma) [ n^4+6*n^3+14*n^2+15*n+6: n in [0..32] ];
%o (PARI) a(n)=n^4+6*n^3+14*n^2+15*n+6 \\ _Charles R Greathouse IV_, Oct 17 2012
%o (Python)
%o def A176780(n): return n*(n*(n*(n + 6) + 14) + 15) + 6 # _Chai Wah Wu_, Aug 30 2022
%Y Cf. A061804 (2*n*(2*n^2+1)), A082986, A006325 (n*(n-1)*(n^2-n+1)/6), A176711, A176712.
%Y See A169938 for another version.
%K nonn,easy
%O 0,1
%A _Klaus Brockhaus_, Apr 25 2010
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