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A125232 Triangle T(n,k) read by rows: the (n-k)-th term of the k-fold iterated partial sum of the pentagonal numbers.

%I A125232 #25 Mar 25 2019 12:01:37
%S A125232 1,5,1,12,6,1,22,18,7,1,35,40,25,8,1,51,75,65,33,9,1,70,126,140,98,42,
%T A125232 10,1,92,196,266,238,140,52,11,1,117,288,462,504,378,192,63,12,1,145,
%U A125232 405,750,966,882,570,255,75,13,1,176,550,1155,1716,1848,1452,825,330,88,14,1
%N A125232 Triangle T(n,k) read by rows: the (n-k)-th term of the k-fold iterated partial sum of the pentagonal numbers.
%D A125232 Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, p 189.
%H A125232 Robert Israel, <a href="/A125232/b125232.txt">Table of n, a(n) for n = 1..10011</a>(rows 0 to 140, flattened)
%F A125232 T(n,0)=A000326(n). T(n,k)=T(n-1,k) + T(n-1,k-1), k>0. - _R. J. Mathar_, Jun 09 2008
%F A125232 G.f. as triangle: (1+2*x)/((1-x)^2*(1-x-x*y)). - _Robert Israel_, Nov 07 2016
%e A125232 First few rows of the triangle are:
%e A125232    1;
%e A125232    5,   1;
%e A125232   12,   6,   1;
%e A125232   22,  18,   7,   1;
%e A125232   35,  40,  25,   8,   1;
%e A125232   51,  75,  65,  33,   9,   1;
%e A125232   70, 126, 140,  98,  42,  10,   1;
%e A125232   ...
%e A125232 Example: (5,3) = 65 = 25 + 40 = (4,3) + (4,2).
%p A125232 A125232 := proc(n,k) option remember ; if k = 0 then A000326(n) ; elif k = n-1 then 1 ; else procname(n-1,k)+procname(n-1,k-1) ; fi : end: # _R. J. Mathar_, Jun 09 2008
%t A125232 nmax = 11; col[1] = Table[n(3n-1)/2, {n, 1, nmax}]; col[k_] := col[k] = Prepend[Accumulate[col[k-1]], 0]; Table[col[k][[n]], {n, 1, nmax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 25 2019 *)
%Y A125232 Columns: A000326 (pentagonal numbers), A002411, A001296, A051836, A051923.
%Y A125232 Cf. A095264 (row sums).
%K A125232 nonn,tabl
%O A125232 1,2
%A A125232 _Gary W. Adamson_, Nov 24 2006
%E A125232 Edited and extended by _R. J. Mathar_, Jun 09 2008