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A191358 Sequencing of all multinomial coefficients arranged in a s*r array of Pascal simplices P(s,r) and sequenced along the array's anti-diagonals. Each P(s,r) is, in turn, a sequence of terms representing the coefficients of a_1,...,a_s in the expansion of (Sum(a_i, i=1, s))^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 anti-diagonals 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[Binomial[s+r-1,p],{p,0,s-2}].

LINKS

Table of n, a(n) for n=1..348.

Wikipedia, Pascal's simplex.

FORMULA

The Pascal simplex P(s,r) starts at a(n) where n=2^(s+r-1)+Sum[Binomial[s+r-1,p],{p,0,s-2}]. 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

  anti-diagonals 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: A231071 A209156 A191004 * A204133 A062378 A073753

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 August 27 09:13 EDT 2014. Contains 246133 sequences.