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A211357 Triangle read by rows: T(n,k) is the number of noncrossing partitions up to rotation of an n-set that contain k singleton blocks. 4
1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 2, 3, 2, 2, 0, 1, 5, 6, 9, 4, 3, 0, 1, 6, 15, 18, 15, 5, 3, 0, 1, 15, 36, 56, 42, 29, 7, 4, 0, 1, 28, 91, 144, 142, 84, 42, 10, 4, 0, 1, 67, 232, 419, 432, 322, 152, 66, 12, 5, 0, 1, 145, 603, 1160, 1365, 1080, 630, 252, 90, 15, 5, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,11

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..90 from Tilman Piesk)

FORMULA

T(n,k) = (1/n)*(A091867(n,k) - A171128(n,k) + Sum_{d|gcd(n,k)} phi(d) * A171128(n/d, k/d)) for n > 0. - Andrew Howroyd, Nov 16 2017

EXAMPLE

From Andrew Howroyd, Nov 16 2017: (Start)

Triangle begins: (n >= 0, 0 <= k <= n)

   1;

   0,   1;

   1,   0,   1;

   1,   1,   0,   1;

   2,   1,   2,   0,   1;

   2,   3,   2,   2,   0,   1;

   5,   6,   9,   4,   3,   0,  1;

   6,  15,  18,  15,   5,   3,  0,  1;

  15,  36,  56,  42,  29,   7,  4,  0, 1;

  28,  91, 144, 142,  84,  42, 10,  4, 0, 1;

  67, 232, 419, 432, 322, 152, 66, 12, 5, 0, 1;

(End)

MATHEMATICA

a91867[n_, k_] := If[k == n, 1, (Binomial[n + 1, k]/(n + 1)) Sum[Binomial[n + 1 - k, j] Binomial[n - k - j - 1, j - 1], {j, 1, (n - k)/2}]];

a2426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}];

a171128[n_, k_] := Binomial[n, k]*a2426[n - k];

T[0, 0] = 1;

T[n_, k_] := (1/n)*(a91867[n, k] - a171128[n, k] + Sum[EulerPhi[d]* a171128[n/d, k/d], {d, Divisors[GCD[n, k]]}]);

Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 03 2018, after Andrew Howroyd *)

PROG

(PARI)

g(x, y) = {1/sqrt((1 - (1 + y)*x)^2 - 4*x^2) - 1}

S(n)={my(A=(1-sqrt(1-4*x/(1-(y-1)*x) + O(x^(n+2))))/(2*x)-1); Vec(1+intformal((A + sum(k=2, n, eulerphi(k)*g(x^k + O(x*x^n), y^k)))/x))}

my(v=S(10)); for(n=1, #v, my(p=v[n]); for(k=0, n-1, print1(polcoeff(p, k), ", ")); print) \\ Andrew Howroyd, Nov 16 2017

CROSSREFS

Column k=0 is A295198.

Row sums are A054357.

Cf. A091867 (noncrossing partitions of an n-set with k singleton blocks), A211359 (up to rotations and reflections).

Cf. A171128.

Sequence in context: A218797 A137289 A211359 * A238416 A063574 A144515

Adjacent sequences:  A211354 A211355 A211356 * A211358 A211359 A211360

KEYWORD

nonn,tabl

AUTHOR

Tilman Piesk, Apr 12 2012

STATUS

approved

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Last modified December 16 06:18 EST 2019. Contains 330016 sequences. (Running on oeis4.)