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A177708 Pentagonal triangle.

%I A177708 #15 May 15 2019 04:56:21
%S A177708 1,6,12,18,57,51,40,156,209,145,75,330,531,534,330,126,600,1074,1278,
%T A177708 1122,651,196,987,1895,2488,2559,2081,1162,288,1512,3051,4275,4824,
%U A177708 4563,3537,1926,405,2196,4599,6750,8100,8370,7506,5634,3015
%N A177708 Pentagonal triangle.
%C A177708 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 A177708 T(n,1) = A002411(n).
%F A177708 T(n,2) = n*(n-1)*(7*n-2)/2.
%F A177708 T(n,3) = (n-2)*(19*n^2-26*n+9)/2 = Sum_{i=2n..3(n-1)} A000326(i).
%e A177708 The row for n = 4 is (1+5+12+22), (35+51+70), (92+117), 145 => 40, 156, 209, 145.
%e A177708     1;
%e A177708     6,   12;
%e A177708    18,   57,   51;
%e A177708    40,  156,  209,   145;
%e A177708    75,  330,  531,   534,   330;
%e A177708   126,  600, 1074,  1278,  1122,   651;
%e A177708   196,  987, 1895,  2488,  2559,  2081,  1162;
%e A177708   288, 1512, 3051,  4275,  4824,  4563,  3537,  1926;
%e A177708   405, 2196, 4599,  6750,  8100,  8370,  7506,  5634, 3015;
%e A177708   550, 3060, 6596, 10024, 12570, 13775, 13450, 11631, 8534, 4510;
%p A177708 A000326 :=proc(n) n*(3*n-1)/2 ; end proc:
%p A177708 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 A177708 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 A177708 Cf. A000217, A000326, A002411, A093445, A236770 (right border).
%K A177708 nonn,easy,tabl
%O A177708 1,2
%A A177708 _Jonathan Vos Post_, Dec 11 2010