A220901 - OEIS

%I #21 Oct 06 2018 06:42:08

%S 1,1,1,1,1,2,1,3,2,1,4,8,1,5,20,16,1,6,40,96,1,7,70,336,272,1,8,112,

%T 896,2176,1,9,168,2016,9792,7936,1,10,240,4032,32640,79360,1,11,330,

%U 7392,89760,436480,353792,1,12,440,12672,215424,1745920,4245504

%N Triangle read by rows: k-th "a-number" of star graph K_{1,n-1}.

%H Alois P. Heinz, <a href="/A220901/b220901.txt">Rows n = 0..200, flattened</a>

%H Suyoung Choi and Hanchul Park, <a href="http://arxiv.org/abs/1210.3776">A new graph invariant arises in toric topology</a>, arXiv preprint arXiv:1210.3776 [math.AT], 2012-2013. See Table 4.

%F T(n,k) = binomial(n-1, 2*k-1)*A000111(2*k-1) (see Theorem 2.9 in paper). - _Michel Marcus_, Feb 07 2013

%e Triangle begins:

%e   1;

%e   1;

%e   1, 1;

%e   1, 2;

%e   1, 3,  2;

%e   1, 4,  8;

%e   1, 5, 20,  16;

%e   1, 6, 40,  96;

%e   1, 7, 70, 336, 272;

%e   ...

%t t[n_, k_] := t[n, k] = If[k == 0, Boole[n == 0], t[n, k-1] + t[n-1, n-k]];

%t T[n_, k_] := If[k == 0, 1, Binomial[n-1, 2k-1] t[2k-1, 2k-1]];

%t Table[T[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten (* _Jean-François Alcover_, Oct 06 2018 *)

%o (PARI) T(n,k) = {if (k == 0, return(1)); binomial(n - 1, 2*k - 1)*(2*k - 1)!*polcoeff(tan(x + O(x^(2*n + 2))), 2*k - 1);} \\ _Michel Marcus_, Feb 07 2013

%K nonn,tabf

%O 0,6

%A _N. J. A. Sloane_, Jan 01 2013

%E More terms from _Michel Marcus_, Feb 07 2013