A218826 -id:A218826 - cp's OEIS frontend

A239894 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 slices.

1, 1, 1, 4, 2, 1, 25, 9, 3, 1, 217, 58, 15, 4, 1, 2470, 500, 100, 22, 5, 1, 35647, 5574, 861, 152, 30, 6, 1, 637129, 78595, 9435, 1313, 215, 39, 7, 1, 13843948, 1376162, 130159, 14192, 1870, 290, 49, 8, 1, 360022957, 29417919, 2232792, 191850, 20001, 2547, 378, 60, 9, 1

Offset: 1

N. J. A. Sloane, Apr 04 2014

			Triangle begins:
1
1 1
4 2 1
25 9 3 1
217 58 15 4 1
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Kreweras, G.; 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 A218826.

\\ here G(n) is A000366(n).
G(n)={(-1/2)^(n-2)*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1))}
A(n)={my(M=matrix(n,n)); for(n=1, n, for(k=2, n, M[n,k] = sum(i=1, n-k+1, M[i,1]*M[n-i, k-1])); M[n,1]=G(n+1)-sum(i=2, n, M[n,i])); M}
{my(T=A(10)); for(n=1, #T, print(T[n, 1..n]))} \\ Andrew Howroyd, Feb 24 2020

T(n,k) = b(1)*T(n-1,k-1)+b(2)*T(n-2,k-1)+...+b(n-k+1)*T(k-1,k-1), where b(i) = A218826(i) for k > 1.

T(n,k) = Sum_{i=1..n-k+1} A218826(i)*T(n-i, k-1) for k > 1. - Andrew Howroyd, Feb 24 2020

Terms a(16) and beyond from Andrew Howroyd, Feb 19 2020

A218827 Number of indecomposable (by shuffling) alternating n-anagrams.

Original entry on oeis.org

1, 1, 3, 16, 129, 1438, 20955, 384226, 8623101, 231978454, 7359117591, 271673905642, 11543742745689, 559348370431630, 30659822500574739, 1887796293833267746, 129757032076160998677, 9900820197631733600518, 834421415151529202479023, 77318409826165250051727514

Offset: 1

Michel Marcus, Nov 07 2012

nonn

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

For n>2, a(n) are alternatively congruent to 3 and -2 modulo 18.

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, A218826. First column of A239895.

m = 20(*terms*); matc = Array[0&, {m, m}];
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]]]];
a[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]];
Array[a, m] (* Jean-François Alcover, Aug 03 2019, from PARI *)

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

cp's OEIS Frontend

A239894 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 slices.

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