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 A127743 #37 May 05 2025 23:40:55
%S A127743 1,1,1,2,2,1,6,5,3,1,22,16,9,4,1,92,60,31,14,5,1,426,252,120,52,20,6,
%T A127743 1,2146,1160,510,209,80,27,7,1,11624,5776,2348,904,335,116,35,8,1,
%U A127743 67146,30832,11610,4184,1481,507,161,44,9,1
%N A127743 Triangular array where T(n,k) is the number of set partitions of n with k atomic parts.
%C A127743 Triangular array distributing the Bell numbers (A000110). The value associated with each partition is the product of A074664(k) for each part of size k, times the number of compositions associated with the partition (A048996 & A072881). The value for T(n,k) is the total of these values for each partition of n into k parts.
%C A127743 Calculating the appropriate weights can be done by "working backward". Suppose for example we know the weights for 1 through 6 and desire the weight for the partitions of seven: Substitute the weights for each partition value and multiply. For example, 7 = 4+3 so f([4,3]) = 6*2 = 12; adjusting for the number of permutations of [4,3] we now have 2*12 = 24. Continuing in this manner for each partition of seven and summing to 451 we now know all of the values except that associated with the partition [7] which must be 877 - 451 = 426.
%C A127743 From _Mike Zabrocki_: (Start)
%C A127743 Every set partition can be uniquely split into "atomic" set partitions or is itself already atomic.
%C A127743 {{1},{2},{3}} = {{1}}|{{1}}|{{1}}
%C A127743 {{1},{23}} = {{1}}|{{12}}
%C A127743 {{12},{3}} = {{12}}|{{1}}
%C A127743 {{13},{2}} is already atomic
%C A127743 {{123}} is already atomic
%C A127743 where this operation | is defined as {A1,...,Ar}|{B1,...,Bs} = {A1,...,Ar,B1+n,...,Bs+n}
%C A127743 where Bi+n = {bi1+n,bi2+n,...,bik+n} if Bi = {bi1,bi2,...,bik} and n = |A1|+|A2|+...+|Ar|. (End)
%C A127743 Subtriangle (n >= 1 and 1 <= k <= n) of triangle given by [0,1,1,2,1,3,1,4,1,5,1,6,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 03 2007
%C A127743 From _Peter Bala_, Aug 05 2014: (Start)
%C A127743 Let B(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + ... denote the o.g.f. for the Bell numbers A000110. Let f(x) = (B(x) - 1)/(x*B(x)) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + ..., the o.g.f. for the first column of this array. Then this array appears to be the Riordan array (f(x), x*f(x)).
%C A127743 If true, this gives the o.g.f. of the array as (B(x) - 1)/( x*(t + (1 - t)*B(x)) ) = 1 + (1 + t)*x + (2 + 2*t + t^2)*x^2 + ... and also the hockey-stick recurrence: T(n+1,k+1) = T(n,k) + T(n-1,k) + 2*T(n-2,k) + 6*T(n-3,k) + 22*T(n-4,k) + ..., n,k >= 1. (End)
%H A127743 G. C. Greubel, <a href="/A127743/b127743.txt">Rows n=1..100 of triangle, flattened</a>
%F A127743 T(n, m) = Sum_{k=1..n-m}( Sum_{i=1..n-m-k}(T(k+i, k)*C(n-m-k-1, n-m-k-i))*C(k+m-1, k) ) + C(n-1, n-m). - _Vladimir Kruchinin_, Mar 21 2015
%e A127743 The partitions of 4 are
%e A127743 4 31 22 211 1111
%e A127743 and the products are
%e A127743 1*6 2*2 1*1 3*1 1*1
%e A127743 therefore row 4 of the table is
%e A127743 6 5 3 1.
%e A127743 From _Philippe Deléham_, Aug 03 2007: (Start)
%e A127743 Triangle begins:
%e A127743 1;
%e A127743 1, 1;
%e A127743 2, 2, 1;
%e A127743 6, 5, 3, 1;
%e A127743 22, 16, 9, 4, 1;
%e A127743 92, 60, 31, 14, 5, 1;
%e A127743 426, 252, 120, 52, 20, 6, 1;
%e A127743 2146, 1160, 510, 209, 80, 27, 7, 1; ...
%e A127743 Triangle [0,1,1,2,1,3,1,4,1,...] DELTA [1,0,0,0,0,0,...] begins:
%e A127743 1;
%e A127743 0, 1;
%e A127743 0, 1, 1;
%e A127743 0, 2, 2, 1;
%e A127743 0, 6, 5, 3, 1;
%e A127743 0, 22, 16, 9, 4, 1;
%e A127743 0, 92, 60, 31, 14, 5, 1;
%e A127743 0, 426, 252, 120, 52, 20, 6, 1;
%e A127743 0, 2146, 1160, 510, 209, 80, 27, 7, 1; ...
%e A127743 (End)
%t A127743 T[n_, m_] := T[n, m] = Sum[Sum[T[k+i, k]*Binomial[n-m-k-1, n-m-k-i], {i, 1, n-m-k}]*Binomial[k+m-1, k], {k, 1, n-m}] + Binomial[n-1, n-m]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 23 2015, after _Vladimir Kruchinin_ *)
%o A127743 (Maxima)
%o A127743 T(n,m):=sum((sum(T(k+i,k)*binomial(n-m-k-1,n-m-k-i),i,1,n-m-k))*binomial(k+m-1,k),k,1,n-m)+binomial(n-1,n-m); /* _Vladimir Kruchinin_, Mar 21 2015 */
%o A127743 (PARI) {T(n,m) = sum(k=1,n-m, (sum(i=1, n-m-k, (T(k+i, k)*binomial(n-m-k-1, n-m-k-i))*binomial(k+m-1, k)))) + binomial(n-1, n-m)};
%o A127743 for(n=1, 10, for(m=1, n, print1(T(n,m), ", "))) \\ _G. C. Greubel_, Dec 06 2018
%Y A127743 Cf. A000041, A000110 (row sums), A074664 (1st column), A048996, A072881, A036043, A036042, A084938.
%K A127743 nonn,tabl
%O A127743 1,4
%A A127743 _Alford Arnold_, Feb 24 2007
%E A127743 Edited by _Franklin T. Adams-Watters_, Jan 25 2010