login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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