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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
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, 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, 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
OFFSET
1,10
COMMENTS
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).
FORMULA
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)).
EXAMPLE
The Pascal simplex P(4,5) for the coefficients of (a_1 + a_2 + a_3 + a_4)^5 is the sequence:
.......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
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).
Within the Pascal simplex P(4,5) the term S(5,3,2,1) = binomial(5,3)*binomial(3,2)*binomial(2,1) = 60.
MATHEMATICA
p[s_, r_] := (f[t_] := Binomial[k[t - 1], k[t]] f[t - 1]; f[1] = 1;
dim = s; k[1] = r; list = {}; vstring[0] = "{k[``], 0, k[``]}, ";
Do[vstring[i] = ToString[StringForm[vstring[0], i + 1, i]], {i, 1, dim - 1}];
dostring = "Do[AppendTo[list, f[dim]], ]";
Do[dostring =
StringInsert[dostring, vstring[j], StringLength[dostring]], {j, dim - 1}];
dostring = StringDrop[dostring, {StringLength[dostring] - 1}];
ToExpression[dostring];
Flatten[List[list]])
g[m_] := (For[h = 1; c = 1, c > 0, h++, c = m - h (h + 1)/2;
a = m - h (h - 1)/2]; b = h - 1 - a; p[a, b])
Flatten[Table[g[e], {e, 1, 40}]]
CROSSREFS
Sequence in context: A329325 A341149 A191004 * A204133 A342017 A062378
KEYWORD
nonn,tabf,easy
AUTHOR
Frank M Jackson, May 31 2011
STATUS
approved