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A209612 Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations and reflections. 6
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 8, 8, 3, 1, 1, 3, 12, 17, 12, 3, 1, 1, 4, 19, 41, 41, 19, 4, 1, 1, 4, 27, 78, 116, 78, 27, 4, 1, 1, 5, 38, 148, 298, 298, 148, 38, 5, 1, 1, 5, 50, 250, 680, 932, 680, 250, 50, 5, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Like the Narayana triangle A001263 (and unlike A152176) this triangle is symmetric.

LINKS

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

Tilman Piesk, Partition related number triangles

FORMULA

T(n,k) = (A088855(n,k) + A209805(n,k))/2. - Andrew Howroyd, Nov 15 2017

EXAMPLE

Triangle begins:

1;

1,  1;

1,  1,  1;

1,  2,  2,  1;

1,  2,  4,  2,  1;

1,  3,  8,  8,  3,  1;

1,  3, 12, 17, 12,  3,  1;

1,  4, 19, 41, 41, 19,  4,  1;

1,  4, 27, 78,116, 78, 27,  4,  1;

1,  5, 38,148,298,298,148, 38,  5,  1

MATHEMATICA

b[n_, k_] := Binomial[n - 1, n - k]*Binomial[n, n - k];

T[n_, k_] := (n*Binomial[Quotient[n - 1, 2], Quotient[k - 1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]] + DivisorSum[GCD[n, k], EulerPhi[#]* b[n/#, k/#]&] + DivisorSum[GCD[n, k - 1], EulerPhi[#]*b[n/#, (n + 1 - k)/#]&] - k*Binomial[n, k]^2/(n - k + 1))/(2*n);

Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 30 2018, after Andrew Howroyd *)

PROG

(PARI)

b(n, k)=binomial(n-1, n-k)*binomial(n, n-k);

T(n, k)=(n*binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2) + sumdiv(gcd(n, k), d, eulerphi(d)*b(n/d, k/d)) + sumdiv(gcd(n, k-1), d, eulerphi(d)*b(n/d, (n+1-k)/d)) - k*binomial(n, k)^2/(n-k+1))/(2*n); \\ Andrew Howroyd, Nov 15 2017

CROSSREFS

Cf. A111275 (row sums)

Cf. A088855, A209805.

Sequence in context: A172453 A172479 A122085 * A209805 A238453 A066287

Adjacent sequences:  A209609 A209610 A209611 * A209613 A209614 A209615

KEYWORD

nonn,tabl

AUTHOR

Tilman Piesk, Mar 10 2012

STATUS

approved

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Last modified December 10 15:09 EST 2019. Contains 329896 sequences. (Running on oeis4.)