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%I #35 Feb 08 2022 20:02:07
%S 1,8,32,90,205,406,728,1212,1905,2860,4136,5798,7917,10570,13840,
%T 17816,22593,28272,34960,42770,51821,62238,74152,87700,103025,120276,
%U 139608,161182,185165,211730,241056,273328,308737,347480,389760,435786,485773,539942,598520
%N Binomial transform of [1, 7, 17, 17, 6, 0, 0, 0, ...].
%C Equals row sums of triangle A178067. - _Gary W. Adamson_, May 18 2010
%C Antidiagonal sums of the convolution array A213771. - _Clark Kimberling_, Jul 04 2012
%C Partial sums of A081436. - _J. M. Bergot_, Jun 20 2013
%H Vincenzo Librandi, <a href="/A132117/b132117.txt">Table of n, a(n) for n = 1..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 Let M = the infinite lower triangular matrix of the natural numbers: [1; 2,3; 4,5,6; ...]; and V = [1, 2, 3, ...]. Then M*V = A132117.
%F O.g.f.: -x(1+x)(2x+1)/(-1+x)^5. - _R. J. Mathar_, Apr 02 2008
%F a(n) = (4*n + 6*n^2 + 8*n^3 + 6*n^4)/24. - _Alois P. Heinz_, Aug 07 2008
%F a(n) = A000217(n)^2 - Sum_{i=1..n-1} A000217(i) = n*(n+1)*(3*n^2+n+2)/12. - _Bruno Berselli_, May 01 2010
%e a(3) = 32 = (1, 2, 1) dot (1, 7, 17) = (1 + 14 + 17).
%e a(5) = 15^2 - (10+6+3+1) = A000537(5) - A000292(4) = 225 - 20 = 205. - _Bruno Berselli_, May 01 2010
%p a:= n-> (Matrix([[0,0,2,13,46]]). Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [5,-10,10,-5,1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..29); # _Alois P. Heinz_, Aug 07 2008
%p a:= n-> (4+(6+(8+6*n)*n)*n)*n/24: seq(a(n),n=1..40); # _Alois P. Heinz_, Aug 07 2008
%t Table[(4 n + 6 n^2 + 8 n^3 + 6 n^4) / 24, {n, 50}] (* _Vincenzo Librandi_, Jun 21 2013 *)
%o (PARI) a(n) = (4*n+6*n^2+8*n^3+6*n^4)/24 \\ _Charles R Greathouse IV_, Sep 03 2011
%Y Cf. A178067. - _Gary W. Adamson_, May 18 2010
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Aug 10 2007
%E More terms from _R. J. Mathar_, Apr 02 2008
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