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A303929
Array read by antidiagonals: T(n,k) is the number of noncrossing partitions up to rotation and reflection composed of n blocks of size k.
8
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 6, 1, 1, 1, 1, 3, 8, 13, 12, 1, 1, 1, 1, 4, 11, 34, 49, 27, 1, 1, 1, 1, 4, 16, 60, 169, 201, 65, 1, 1, 1, 1, 5, 20, 109, 423, 1019, 940, 175, 1, 1, 1, 1, 5, 26, 167, 918, 3381, 6710, 4643, 490, 1
OFFSET
0,14
LINKS
EXAMPLE
=================================================================
n\k| 1 2 3 4 5 6 7 8 9
---+-------------------------------------------------------------
0 | 1 1 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 1 1 ...
2 | 1 1 1 1 1 1 1 1 1 ...
3 | 1 2 2 3 3 4 4 5 5 ...
4 | 1 3 5 8 11 16 20 26 32 ...
5 | 1 6 13 34 60 109 167 257 359 ...
6 | 1 12 49 169 423 918 1741 3051 4969 ...
7 | 1 27 201 1019 3381 9088 20569 41769 77427 ...
8 | 1 65 940 6710 29335 96315 259431 607696 1280045 ...
9 | 1 175 4643 47104 266703 1072187 3417520 9240444 22066742 ...
...
MATHEMATICA
u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #]&] + DivisorSum[GCD[n - 1, k], EulerPhi[#]*Binomial[n*k/#, (n - 1)/#]&])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
Table[T[n - k, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 14 2018, translated from PARI *)
PROG
(PARI) \\ here c(n, k) is A303694
u(n, k, r) = {r*binomial(k*n + r, n)/(k*n + r)}
e(n, k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
c(n, k)={if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(k*d, d)) + sumdiv(gcd(n-1, k), d, eulerphi(d)*binomial(n*k/d, (n-1)/d)))/(k*n) - binomial(k*n, n)/(n*(k-1)+1))}
T(n, k)={(1/2)*(c(n, k) + if(n==0, 1, if(k%2, if(n%2, 2*u(n\2, k, (k+1)/2), u(n/2, k, 1) + u(n/2-1, k, k)), e(n, k) + if(n%2, u(n\2, k, k/2)))/2))}
CROSSREFS
Columns 2..5 are A006082(n+1), A082938, A303870, A303871.
Sequence in context: A321724 A333737 A333733 * A303694 A194673 A240595
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, May 02 2018
STATUS
approved