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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 (list; graph; refs; listen; history; text; internal format)
 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). LINKS Wikipedia, Pascal's simplex. 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: A209156 A329325 A191004 * A204133 A342017 A062378 Adjacent sequences:  A191355 A191356 A191357 * A191359 A191360 A191361 KEYWORD nonn,tabf,easy AUTHOR Frank M Jackson, May 31 2011 STATUS approved

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Last modified April 14 22:47 EDT 2021. Contains 342971 sequences. (Running on oeis4.)