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 A191358 Sequencing of all multinomial coefficients arranged in an s X r array of Pascal simplices P(s,r) and sequenced along the array's antidiagonals. Each P(s,r) is, in turn, a sequence of terms representing the coefficients of a_1,...,a_s in the expansion of (Sum_{i=1..s} a_i)^r with r starting at zero. 0

%I

%S 1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,3,1,1,2,2,1,2,1,1,1,1,1,1,1,1,4,

%T 6,4,1,1,3,3,3,6,3,1,3,3,1,1,2,2,2,1,2,2,1,2,1,1,1,1,1,1,1,1,1,5,10,

%U 10,5,1,1,4,4,6,12,6,4,12,12,4,1,4,6,4,1,1,3,3,3,3,6,6,3,6,3,1,3,3,3,6,3,1,3,3,1,1,2,2,2,2,1,2,2,2,1,2,2,1,2,1,1,1,1,1,1,1,1,1,1,6,15,20,15,6,1,1,5,5,10,20,10,10,30,30,10,5,20,30,20,5,1,5,10,10,5,1,1,4,4,4,6,12,12,6,12,6,4,12,12,12,24,12,4,12,12,4,1,4,4,6,12,6,4,12,12,4,1,4,6,4,1,1,3,3,3,3,3,6,6,6,3,6,6,3,6,3,1,3,3,3,3,6,6,3,6,3,1,3,3,3,6,3,1,3,3,1,1,2,2,2,2,2,1,2,2,2,2,1,2,2,2,1,2,2,1,2,1,1,1,1,1,1,1,1,1,1,1,7,21,35,35,21,7,1,1,6,6,15,30,15,20,60,60,20,15,60,90,60,15,6,30,60,60,30,6,1,6,15,20,15,6,1,1,5,5,5,10,20,20,10,20,10,10,30,30,30,60,30,10,30,30,10,5,20,20,30,60,30,20,60,60,20,5,20,30,20,5,1,5,5,10,20,10,10,30,30,10,5,20,30,20,5,1,5,10,10,5,1

%N Sequencing of all multinomial coefficients arranged in an s X r array of Pascal simplices P(s,r) and sequenced along the array's antidiagonals. Each P(s,r) is, in turn, a sequence of terms representing the coefficients of a_1,...,a_s in the expansion of (Sum_{i=1..s} a_i)^r with r starting at zero.

%C The Pascal simplices P(s,r) are sequenced along the s*r array's antidiagonals as P(1,0), P(1,1), P(2,0), P(1,2), P(2,1), P(3,0), P(1,3), P(2,2), P(3,1), P(4,0), etc. P(2,3) is the sequence 1,3,3,1. P(2,r) = Pascal's triangle = A007318. P(3,r) = Pascal's tetrahedron = A046816. P(4,r) = Pascal's 4D simplex = A189225. Each P(s,r) has binomial(s-1+r, s-1) terms. The sum of its terms is s^r. The Pascal simplex P(s,r) starts at a(n) where n = 2^(s+r-1) + Sum_{p=0..s-2} binomial(s+r-1,p).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pascal%27s_simplex">Pascal's simplex</a>.

%F The Pascal simplex P(s,r) starts at a(n) where n = 2^(s+r-1) + Sum_{p=0..s-2} binomial(s+r-1,p). The individual terms within the Pascal simplex, S(r,t_1,t_2,...,t_(s-1)) are given by S(r,t_1,t_2,...,t_(s-1)) = binomial(r,t_1)*binomial(t_1,t_2)*...*binomial(t_(s-2),t_(s-1)).

%e The Pascal simplex P(4,5) for the coefficients of (a_1 + a_2 + a_3 + a_4)^5 is the sequence:

%e .......1

%e .......5

%e ......5,5

%e .......10

%e .....20,20

%e ....10,20,10

%e .......10

%e .....30,30

%e ....30,60,30

%e ..10,30,30,10

%e .......5

%e .....20,20

%e ....30,60,30

%e ..20,60,60,20

%e ..5 ,20,30,20,5

%e .......1

%e ......5,5

%e ....10,20,10

%e ..10,30,30,10

%e .5, 20,30,20,5

%e 1,5, 10,10, 5,1

%e The sequence starts at a(293), it has 56 terms and the sum of its terms is 1024. It is also the 40th Pascal simplex in the sequence counting along the antidiagonals of the s*r array of Pascal simpices P(s,r).

%e Within the Pascal simplex P(4,5) the term S(5,3,2,1) = binomial(5,3)*binomial(3,2)*binomial(2,1) = 60.

%t p[s_, r_] := (f[t_] := Binomial[k[t - 1], k[t]] f[t - 1]; f[1] = 1;

%t dim = s; k[1] = r; list = {}; vstring[0] = "{k[``],0,k[``]},";

%t Do[vstring[i] = ToString[StringForm[vstring[0], i + 1, i]], {i, 1, dim - 1}];

%t dostring = "Do[AppendTo[list,f[dim]],]";

%t Do[dostring =

%t StringInsert[dostring, vstring[j], StringLength[dostring]], {j, dim - 1}];

%t dostring = StringDrop[dostring, {StringLength[dostring] - 1}];

%t ToExpression[dostring];

%t Flatten[List[list]])

%t g[m_] := (For[h = 1; c = 1, c > 0, h++, c = m - h (h + 1)/2;

%t a = m - h (h - 1)/2]; b = h - 1 - a; p[a, b])

%t Flatten[Table[g[e], {e, 1, 40}]]

%K nonn,tabf,easy

%O 1,10

%A _Frank M Jackson_, May 31 2011

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