This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A191358 #13 Jul 06 2025 11:02:05
%S A191358 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 A191358 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 A191358 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 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.
%C A191358 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 A191358 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pascal%27s_simplex">Pascal's simplex</a>.
%F A191358 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 A191358 The Pascal simplex P(4,5) for the coefficients of (a_1 + a_2 + a_3 + a_4)^5 is the sequence:
%e A191358 .......1
%e A191358 .......5
%e A191358 ......5,5
%e A191358 .......10
%e A191358 .....20,20
%e A191358 ....10,20,10
%e A191358 .......10
%e A191358 .....30,30
%e A191358 ....30,60,30
%e A191358 ..10,30,30,10
%e A191358 .......5
%e A191358 .....20,20
%e A191358 ....30,60,30
%e A191358 ..20,60,60,20
%e A191358 ..5 ,20,30,20,5
%e A191358 .......1
%e A191358 ......5,5
%e A191358 ....10,20,10
%e A191358 ..10,30,30,10
%e A191358 .5, 20,30,20,5
%e A191358 1,5, 10,10, 5,1
%e A191358 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 A191358 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 A191358 p[s_, r_] := (f[t_] := Binomial[k[t - 1], k[t]] f[t - 1]; f[1] = 1;
%t A191358 dim = s; k[1] = r; list = {}; vstring[0] = "{k[``],0,k[``]},";
%t A191358 Do[vstring[i] = ToString[StringForm[vstring[0], i + 1, i]], {i, 1, dim - 1}];
%t A191358 dostring = "Do[AppendTo[list,f[dim]],]";
%t A191358 Do[dostring =
%t A191358 StringInsert[dostring, vstring[j], StringLength[dostring]], {j, dim - 1}];
%t A191358 dostring = StringDrop[dostring, {StringLength[dostring] - 1}];
%t A191358 ToExpression[StringReverse@StringReplace[StringReverse@dostring, ","->"", 1]];
%t A191358 Flatten[List[list]])
%t A191358 g[m_] := (For[h = 1; c = 1, c > 0, h++, c = m - h (h + 1)/2;
%t A191358 a = m - h (h - 1)/2]; b = h - 1 - a; p[a, b])
%t A191358 Flatten[Table[g[e], {e, 1, 40}]]
%K A191358 nonn,tabf,easy
%O A191358 1,10
%A A191358 _Frank M Jackson_, May 31 2011