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

1,3

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Jose Bastidas, Félix Gélinas, Vincent Pilaud, Germain Poullot, Andrew Sack, and Eleni Tzanaki, Interval hypergraphic polytopes (or deformed associahedra), Tamari interval posets, and weeping willows, arXiv:2606.18376 [math.CO], 2026. See p. 16.

Germain Kreweras and Jacques Barraud, Anagrammes alternés, Europ. J. Comb. 18(8) (November 1997), 887-891.

Germain Kreweras and Dominique Dumont, Sur les anagrammes alternés, Disc. Math. 211(1-3) (28 January 2000), 103-110.

Formula

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

Prog

(PARI) 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), ", "); ); }

Crossrefs

Cf. A000366, A218827. First column of A239894.

Sequence in context: A198198 A007830 A305404 * A060911 A060912 A209319

Adjacent sequences: A218823 A218824 A218825 * A218827 A218828 A218829

Keyword

nonn,changed

Author

Michel Marcus, Nov 07 2012

Status

approved