A239895 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = number of alternating anagrams on n letters (of length 2n) which are decomposable into at most k components.

1, 1, 1, 3, 3, 1, 16, 15, 6, 1, 129, 110, 45, 10, 1, 1438, 1104, 435, 105, 15, 1, 20955, 14455, 5334, 1295, 210, 21, 1, 384226, 238536, 81256, 19089, 3220, 378, 28, 1, 8623101, 4834854, 1509246, 335496, 56259, 7056, 630, 36, 1

Offset

1,4

Comments

The Bell transform of A218827(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 17 2016

Links

Table of n, a(n) for n=1..45.

Kreweras, G. and Dumont, D., Sur les anagrammes alternés. (French) [On alternating anagrams] Discrete Math. 211 (2000), no. 1-3, 103--110. MR1735352 (2000h:05013).

Formula

T(n,k) = C(n-1,0)*c(1)*T(n-1,k-1) + C(n-1,1)*c(2)*T(n-2,k-1) + ... + C(n-1,n-1)*c(n-k+1)*T(k-1,k-1), where c(i) = A218827(i).

Example

Triangle begins:
       1;
       1,      1;
       3,      3,     1;
      16,     15,     6,     1;
     129,    110,    45,    10,    1;
    1438,   1104,   435,   105,   15,   1;
   20955,  14455,  5334,  1295,  210,  21,  1;
  384226, 238536, 81256, 19089, 3220, 378, 28, 1;

Mathematica

m = 10(*terms of A218827 for m-1 rows*); matc = Array[0&, {m, m}];
(* The function BellMatrix is defined in A264428.*)
a366[n_] := (-2^(-1))^(n - 2)*Sum[Binomial[n, k]*(1 - 2^(n + k + 1))* BernoulliB[n + k + 1], {k, 0, n}];
ci[n_, k_] := ci[n, k] = Module[{v}, If[matc[[n, k]] == 0, If[n == k, v = 1, If[k == 1, v = c[n], v = Sum[Binomial[n - 1, i - 1]*c[i]*ci[n - i, k - 1], {i, 1, n - k + 1}]]]; matc[[n, k]] = v]; Return[matc[[n, k]] ]];
c[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]]
T = Rest /@ BellMatrix[c[# + 1]&, m] // Rest;
Table[T[[n, k]], {n, 1, m - 1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 03 2019 *)

Prog

(Sage) # uses[bell_matrix from A264428, A218827]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
A239895_generator = lambda n: A218827(n+1)
bell_matrix(A239895_generator, 9) # Peter Luschny, Jan 17 2016

Crossrefs

Row sums are A000366. First column is A218827.

Sequence in context: A123244 A364505 A105599 * A106210 A033842 A104417

Adjacent sequences: A239892 A239893 A239894 * A239896 A239897 A239898

Keyword

nonn,tabl,more

Author

N. J. A. Sloane, Apr 04 2014

Extensions

More terms from Peter Luschny, Jan 17 2016

Status

approved