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A293595 Triangle read by rows: T(n,k) = number of compositions of n into k parts such that no two cyclically adjacent parts are equal. 3
1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 6, 2, 0, 0, 1, 6, 6, 4, 0, 0, 0, 1, 6, 12, 10, 0, 0, 0, 0, 1, 8, 18, 16, 10, 2, 0, 0, 0, 1, 8, 24, 40, 20, 6, 0, 0, 0, 0, 1, 10, 30, 52, 50, 18, 0, 0, 0, 0, 0, 1, 10, 42, 84, 90, 50, 14, 2, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Compositions of length 1 are included.

See theorem 4 in Hadjicostas reference for generating function.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

P. Hadjicostas, Cyclic, dihedral and symmetric Carlitz compositions of a positive integer, Journal of Integer Sequences, 20 (2017), Article 17.8.5.

FORMULA

G.f.: (Sum_{j>=1} x^(2*j)*y^2/(1+x^j*y)) + (Sum_{j>=1} x^j*y/(1+x^j*y)^2) / (1 - Sum_{j>=1} x^j*y/(1+x^j*y)).

EXAMPLE

Triangle begins:

  1;

  1,  0;

  1,  2,  0;

  1,  2,  0,  0;

  1,  4,  0,  0,  0;

  1,  4,  6,  2,  0,  0;

  1,  6,  6,  4,  0,  0,  0;

  1,  6, 12, 10,  0,  0,  0,  0;

  1,  8, 18, 16, 10,  2,  0,  0,  0;

  1,  8, 24, 40, 20,  6,  0,  0,  0,  0;

  ...

Case n=6:

The included compositions are:

k=1: 6;                                => T(6,1) = 1

k=2: 15, 24, 42, 51;                   => T(6,2) = 4

k=3: 123, 132, 213, 231, 312, 321;     => T(6,3) = 6

k=4: 1212, 2121;                       => T(6,4) = 2

MATHEMATICA

max = 10; gf = Sum[x^(2*j)*y^2/(1 + x^j*y), {j, 1, max}] + Sum[x^j*y/(1 + x^j*y)^2, {j, 1, max}]/(1 - Sum[ x^j*y/(1 + x^j*y), {j, 1, max}]) + O[x]^(max+1) + O[y]^(max+1) // Normal // Expand;

T[n_, k_] := SeriesCoefficient[gf, {x, 0, n}, {y, 0, k}];

Table[T[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 19 2018 *)

PROG

(PARI)

gf(n, y) = {my(A=sum(j=1, n, x^(2*j)*y^2/(1+x^j*y) + O(x*x^n)),

B=sum(j=1, n, x^j*y/(1+x^j*y)^2 + O(x*x^n)),

C=sum(j=1, n, x^j*y/(1+x^j*y) + O(x*x^n)));

A + B/(1-C)}

for(n=1, 10, my(p=polcoeff(gf(n, y), n)); for(k=1, n, print1(polcoeff(p, k), ", ")); print)

CROSSREFS

Row sums are in A212322.

Cf. A106351, A106369.

Sequence in context: A079644 A072705 A072574 * A261249 A058650 A112177

Adjacent sequences:  A293592 A293593 A293594 * A293596 A293597 A293598

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Oct 12 2017

STATUS

approved

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Last modified November 21 09:14 EST 2019. Contains 329362 sequences. (Running on oeis4.)