{1},
{0,1},
{0,1,1},
{0,1,1,1},
{0,1,2,1,1},
{0,1,1,3,1,1,},
{0,1,3,1,4,1,1},
{0,1,-1,7,1,5,1,1},
{0,1,15,-17,14,1,6,1,1},
{0,1,-78,133,-61,25,1,7,1,1},
{0,1,632,-1020,529,-152,41,1,8,1,1},
{0,1,-6049,9826,-4989,1506,-314,63,1,9,1,1},
{0,1,68036,-110514,56161,-16668,3532,-576,92,1,10,1,1},
{0,1,-878337,1427046,-724881,214528,-44703,7276,-972,129,1,11,1,1},
{0,1,12817659,-20827070,10576885,-3123249,647092,-103476,13644,-1541,175,1,12,1,1}}

{0,1,-208803098,339300237,-172288208,50807116,-10485698,1663298,-215207,23815,-2327,231,1,13,1,1},
{0,1,3758607935,-6107891965,3101175261,-913762452,188128602,-29683556,3802577,-412385,39277,-3379,298,1,14,1,1},
{0,1,-74132077726,120470718768,-61163565992,18012156613,-3702614501,582218530,-74095782,7943407,-740528,61863,-4751,377,1,15,1,1},
{0,1,1590647874073,-2584976884923,1312357714574,-386340094901,79334502155,-12446802721,1577216769,-167793527,15442384,-1261272,93787,-6502,469,1,16,1,1}

T(n+7,n)=(-16560+3600*n+17554*n^2+1395*n^3-3185*n^4-1755*n^5-329*n^6)/720
T(n+8,n)=(488880 + 193248*n - 629160*n^2 - 266840*n^3 + 86625*n^4 + 97657*n^5 + 30975*n^6 + 3655*n^7)/5040
T(n+9,n)=(-6491520 - 2912280*n + 6402520*n^2 + 5498640*n^3 - 80129*n^4 - 1526280*n^5 - 721070*n^6 - 147840*n^7 - 11961*n^8)/10080

Conjecture: T(n+k,n) (k-1)! is a polynome in n of degree (k-1) with leading coefficient A000806(k-1)
with A000806 =  -Exp[x] BesselK[n + 1/2, x] Sqrt[2/Pi]/Sqrt[x] = [1,0,1,-5,36,-329,3655,-47844,...]

Extended using extra terms in A119342 et al. provided by Sean A. Irvine (sairvin(AT)xtra.co.nz