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 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, ... 6
 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, 151, 203, 877, 203, 255, 322, 409, 523, 674, 877, 4140, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 21147, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Or, prefix Aitken's array (A011971) with a leading diagonal of zeros and take the differences of each row to get the new triangle. 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 Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ... 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 LINKS Chai Wah Wu, Rows n = 0..50, flattened Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011. Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971.  See Proposition 3. A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227). FORMULA 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 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 T(n,1) = B(n-1); T(n,2) = B(n-2) for n>=2; T(n,k) = Sum_{i=0..k-2}binom(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 EXAMPLE Triangle starts:    1;    1,  1;    2,  1,  2;    5,  2,  3,  5;   15,  5,  7, 10, 15;   52, 15, 20, 27, 37, 52; MAPLE 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 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 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 MATHEMATICA 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 *) PROG (Python) # requires Python 3.2 or higher. from itertools import accumulate A095149_list, blist = [1, 1, 1], [1] for _ in range(2*10**2): ....b = blist[-1] ....blist = list(accumulate([b]+blist)) ....A095149_list += [blist[-1]]+ blist # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014 CROSSREFS Cf. A000110, A005493, A011971, A188919, A271466. Sequence in context: A153206 A144155 A109631 * A218579 A182436 A064192 Adjacent sequences:  A095146 A095147 A095148 * A095150 A095151 A095152 KEYWORD nonn,tabl,easy,nice AUTHOR Gary W. Adamson, May 30 2004 EXTENSIONS Corrected and extended by R. J. Mathar, Feb 05 2007 Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007 STATUS approved

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Last modified November 18 17:49 EST 2018. Contains 317323 sequences. (Running on oeis4.)