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%I #15 May 15 2019 04:56:21
%S 1,6,12,18,57,51,40,156,209,145,75,330,531,534,330,126,600,1074,1278,
%T 1122,651,196,987,1895,2488,2559,2081,1162,288,1512,3051,4275,4824,
%U 4563,3537,1926,405,2196,4599,6750,8100,8370,7506,5634,3015
%N Pentagonal triangle.
%C This is to A093445 as pentagonal numbers A000326 are to triangular numbers A000217. The n-th row of the triangular table begins by considering A000217(n) pentagonal numbers (starting with 1) in order. Now segregate them into n chunks beginning with n members in the first chunk, n-1 members in the second chunk, and so forth. Now sum each chunk. Thus the first term is the sum of first n numbers = n*(3n-1)/2, the second term is the sum of the next n-1 terms (from n+1 to 2n-1), the third term is the sum of the next n-2 terms (2n to 3n-3)... This triangle can be called the pentagonal triangle. The sequence contains the triangle by rows. The first column is A002411 (Pentagonal pyramidal numbers: n^2*(n+1)/2).
%F T(n,1) = A002411(n).
%F T(n,2) = n*(n-1)*(7*n-2)/2.
%F T(n,3) = (n-2)*(19*n^2-26*n+9)/2 = Sum_{i=2n..3(n-1)} A000326(i).
%e The row for n = 4 is (1+5+12+22), (35+51+70), (92+117), 145 => 40, 156, 209, 145.
%e 1;
%e 6, 12;
%e 18, 57, 51;
%e 40, 156, 209, 145;
%e 75, 330, 531, 534, 330;
%e 126, 600, 1074, 1278, 1122, 651;
%e 196, 987, 1895, 2488, 2559, 2081, 1162;
%e 288, 1512, 3051, 4275, 4824, 4563, 3537, 1926;
%e 405, 2196, 4599, 6750, 8100, 8370, 7506, 5634, 3015;
%e 550, 3060, 6596, 10024, 12570, 13775, 13450, 11631, 8534, 4510;
%p A000326 :=proc(n) n*(3*n-1)/2 ; end proc:
%p A177708 := proc(n,k) kc := 1 ; nsk := n ; ns := 1 ; while kc < k do ns := ns+nsk ; kc := kc+1 ; nsk := nsk-1 ; end do: add(A000326(i),i=ns..ns+nsk-1) ; end proc: # _R. J. Mathar_, Dec 14 2010
%t Table[Total/@TakeList[PolygonalNumber[5,Range[60]],Range[n,1,-1]],{n,10}]//Flatten (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Feb 17 2018 *)
%Y Cf. A000217, A000326, A002411, A093445, A236770 (right border).
%K nonn,easy,tabl
%O 1,2
%A _Jonathan Vos Post_, Dec 11 2010
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