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.
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
Offset: 0
Examples
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)
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..90 from Tilman Piesk)
Crossrefs
Programs
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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-François Alcover, Jul 03 2018, after Andrew Howroyd *) -
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
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
Comments