A218827 -id:A218827 - cp's OEIS frontend

A218826 Number of indecomposable (by concatenation) alternating n-anagrams.

1, 1, 4, 25, 217, 2470, 35647, 637129, 13843948, 360022957, 11054457253, 396003680518, 16377463914091, 774714094061221, 41572230979229284, 2512149910125036865, 169831839578092130017, 12769241823369505582150, 1062122471116082751430087, 97264621940872013476357969

Offset: 1

Michel Marcus, Nov 07 2012

nonn

Alternating anagrams enumeration is related to A000366 by a(n) = A000366(n+1).

For all n, a(n) are periodically congruent to 1, 1 and 4 modulo 6.

Andrew Howroyd, Table of n, a(n) for n = 1..200
G. Kreweras and J. Barraud, Anagrammes alternés, European Journal of Combinatorics,Volume 18, Issue 8, November 1997, Pages 887-891.
G. Kreweras and D. Dumont, Sur les anagrammes alternés, Discrete Mathematics, Volume 211, Issues 1-3, 28 January 2000, Pages 103-110.

Cf. A000366, A218827. First column of A239894.

a000366(n)= {return((-1/2)^(n-2)*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1)));}
bi(n,k) = {if (matb[n,k] == 0, if (n==k, v=1, if (k==1, v = b(n),v = sum(i=1, n-k+1, b(i)*bi(n-i,k-1)););); matb[n,k] = v;); return (matb[n,k]);}
b(n) = {if (n==1, return(a000366(n+1)), return(a000366(n+1) - sum(i=2, n, bi(n,i))));}
allb(m) = {matb = matrix(m,m); for (i=1, m, print1(b(i), ", "););}

G.f.: (1-Q(0))/x, where Q(k) = 1 - ((k+1)*(k+2)/2)*x/(1 - ((k+1)*(k+2)/2)*x/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/Q(0), where Q(k) = 1 - ((k+1)*(k+2)/2)*x/(1 - ((k+2)*(k+3)/2)*x/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2013
a(n) = A000366(n + 1) - Sum_{k=2..n} A239894(n, k). - Andrew Howroyd, Feb 25 2020

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.

Original entry on oeis.org

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

N. J. A. Sloane, Apr 04 2014

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

			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;

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).

Row sums are A000366. First column is A218827.

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 *)

# 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

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).

More terms from Peter Luschny, Jan 17 2016

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A218826 Number of indecomposable (by concatenation) alternating n-anagrams.

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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.

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