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 A095149 #88 Sep 19 2023 08:19:33
%S A095149 1,1,1,2,1,2,5,2,3,5,15,5,7,10,15,52,15,20,27,37,52,203,52,67,87,114,
%T A095149 151,203,877,203,255,322,409,523,674,877,4140,877,1080,1335,1657,2066,
%U A095149 2589,3263,4140,21147,4140,5017,6097,7432,9089,11155,13744,17007,21147
%N A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...
%C A095149 Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
%C A095149 With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - _Emeric Deutsch_, Oct 29 2006
%C A095149 Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
%C A095149 The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - _Andrew Baxter_, Jun 13 2011
%C A095149 From _Gus Wiseman_, Aug 11 2020: (Start)
%C A095149 Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
%C A095149 1 2 4 8 16
%C A095149 3 18 12 24 48
%C A095149 5 50 20 40 80
%C A095149 7 54 28 56 112
%C A095149 9 1350 108 72 144
%C A095149 25 540 200 400
%C A095149 27 756 360 432
%C A095149 45 504 720
%C A095149 63 600 1008
%C A095149 75 1400 1200
%C A095149 135 2160
%C A095149 175 2800
%C A095149 189 3024
%C A095149 675 10800
%C A095149 4725 75600
%C A095149 Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
%C A095149 Crossrefs:
%C A095149 A008278 is the version counted by omega A001221.
%C A095149 A336420 is the version counted by Omega A001222.
%C A095149 A006939 lists superprimorials or Chernoff numbers.
%C A095149 A008302 counts divisors of superprimorials by Omega.
%C A095149 A022915 counts permutations of prime indices of superprimorials.
%C A095149 A098859 counts partitions with distinct multiplicities.
%C A095149 A130091 lists numbers with distinct prime multiplicities.
%C A095149 A181796 counts divisors with distinct prime multiplicities.
%C A095149 Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
%C A095149 (End)
%H A095149 Alois P. Heinz, <a href="/A095149/b095149.txt">Rows n = 0..150, flattened</a> (first 51 rows from Chai Wah Wu)
%H A095149 Andrew M. Baxter and Lara K. Pudwell, <a href="http://arxiv.org/abs/1108.2642">Enumeration schemes for dashed patterns</a>, arXiv:1108.2642 [math.CO], 2011.
%H A095149 Anders Claesson, <a href="http://dx.doi.org/10.1006/eujc.2001.0515">Generalized pattern avoidance</a>, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
%H A095149 A. Bernini, M. Bouvel and L. Ferrari, <a href="http://puma.dimai.unifi.it/18_3_4/BerniniBouvelFerrari.pdf">Some statistics on permutations avoiding generalized patterns</a>, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).
%F A095149 With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - _Emeric Deutsch_, Oct 29 2006
%F A095149 T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - _Emeric Deutsch_, Oct 29 2006
%e A095149 Triangle starts:
%e A095149 1;
%e A095149 1, 1;
%e A095149 2, 1, 2;
%e A095149 5, 2, 3, 5;
%e A095149 15, 5, 7, 10, 15;
%e A095149 52, 15, 20, 27, 37, 52;
%e A095149 From _Gus Wiseman_, Aug 11 2020: (Start)
%e A095149 Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
%e A095149 {1}{234} {12}{34} {123}{4} {1234}
%e A095149 {1}{2}{34} {12}{3}{4} {13}{24} {124}{3}
%e A095149 {1}{23}{4} {13}{2}{4} {134}{2}
%e A095149 {1}{24}{3} {14}{23}
%e A095149 {1}{2}{3}{4} {14}{2}{3}
%e A095149 (End)
%p A095149 with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%p A095149 Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - _Emeric Deutsch_, Oct 29 2006
%p A095149 A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # _R. J. Mathar_, Feb 05 2007
%p A095149 # alternative Maple program:
%p A095149 b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
%p A095149 b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
%p A095149 end:
%p A095149 T:= (n, k)-> b(n, 0, n-k):
%p A095149 seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Dec 20 2018
%t A095149 nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* _Jean-François Alcover_, Nov 15 2011, from formula with offset 1 *)
%o A095149 (Python)
%o A095149 # requires Python 3.2 or higher.
%o A095149 from itertools import accumulate
%o A095149 A095149_list, blist = [1,1,1], [1]
%o A095149 for _ in range(2*10**2):
%o A095149 b = blist[-1]
%o A095149 blist = list(accumulate([b]+blist))
%o A095149 A095149_list += [blist[-1]]+ blist
%o A095149 # _Chai Wah Wu_, Sep 02 2014, updated _Chai Wah Wu_, Sep 20 2014
%Y A095149 Cf. A000110, A005493, A008278, A011971, A188919, A271466.
%Y A095149 T(2n,n) gives A020556.
%K A095149 nonn,tabl,easy,nice
%O A095149 0,4
%A A095149 _Gary W. Adamson_, May 30 2004
%E A095149 Corrected and extended by _R. J. Mathar_, Feb 05 2007
%E A095149 Entry revised by _N. J. A. Sloane_, Jun 01 2005, Jun 16 2007