A185588 - OEIS

%I #13 Jul 08 2017 12:44:15

%S 1,1,1,1,3,6,4,1,6,24,76,147,198,108,1,10,64,332,1475,5074,14260,

%T 32464,52032,57600,27648,1,15,139,1027,6610,38124,189255,822489,

%U 3164477,10692485,30443198,72934740,141861200,202056000,197280000,86400000

%N Triangular array read by rows. The n-th row is the expansion of (1+x)(1+2x+4x^2)...(1+nx+(nx)^2+(nx)^3+...(nx)^n).

%C T(n,k) is the sum of products of the elements in the size k submultisets of the multiset {1,2,2,3,3,3,...n} which contains i copies of i, 1<=i<=n.

%C The n-th row has n*(n+1)/2+1 elements: 0 <= k <= A000217(n).

%H G. C. Greubel, <a href="/A185588/b185588.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F O.g.f. for row n: Product_{j=1..n} Sum_{i=0..j} (j*x)^i.

%e T(3,2) = 24 because the size 2 submultisets of {1,2,2,3,3,3} are: {1,2},{1,3}, {2,2}, {2,3}, {3,3}. And 1*2 + 1*3 + 2*2 + 2*3 + 3*3 = 24.

%e Triangle T(n,k) begins:

%e 1;

%e 1,  1;

%e 1,  3,  6,   4;

%e 1,  6, 24,  76,  147,  198,   108;

%e 1, 10, 64, 332, 1475, 5074, 14260, 32464, 52032, 57600, 27648;

%p T:= (n,k)-> coeff (mul (add ((j*x)^i, i=0..j), j=1..n), x, k):

%p seq (seq (T(n,k), k=0..n*(n+1)/2), n=0..7);

%t Table[CoefficientList[Series[Product[Sum[(j x)^i,{i,0,j}],{j,1,n}],{x,0,20}],x],{n,0,5}]//Grid

%Y Cf. A000217.

%K nonn,tabf

%O 0,5

%A _Geoffrey Critzer_, Feb 04 2011