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Triangle T(n,k) read by rows: the (n-k)-th term of the k-fold iterated partial sum of the pentagonal numbers.
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%I #25 Mar 25 2019 12:01:37

%S 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 10,1,92,196,266,238,140,52,11,1,117,288,462,504,378,192,63,12,1,145,

%U 405,750,966,882,570,255,75,13,1,176,550,1155,1716,1848,1452,825,330,88,14,1

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

%D Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, p 189.

%H Robert Israel, <a href="/A125232/b125232.txt">Table of n, a(n) for n = 1..10011</a>(rows 0 to 140, flattened)

%F 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 G.f. as triangle: (1+2*x)/((1-x)^2*(1-x-x*y)). - _Robert Israel_, Nov 07 2016

%e First few rows of the triangle are:

%e 1;

%e 5, 1;

%e 12, 6, 1;

%e 22, 18, 7, 1;

%e 35, 40, 25, 8, 1;

%e 51, 75, 65, 33, 9, 1;

%e 70, 126, 140, 98, 42, 10, 1;

%e ...

%e Example: (5,3) = 65 = 25 + 40 = (4,3) + (4,2).

%p 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 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 Columns: A000326 (pentagonal numbers), A002411, A001296, A051836, A051923.

%Y Cf. A095264 (row sums).

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Nov 24 2006

%E Edited and extended by _R. J. Mathar_, Jun 09 2008