Hanna's triangle A096651

[wouter meeussen]

for those who remember, a lower triangular matrix H such that the row sums of H^k give the count of
k-dimensional partitions. Not the end of math as we know it, but still ... (hmm).

News:
the few regularities of H[n,m] known are:
T(n,n)=1
T(n+1,n)=1
T(n+2,n)=n
T(n+3,n)=1
T(n+4,n)=(0+5*n+0*n^2+n^3)/6
T(n+5,n)=(-48+90*n-7*n^2-6*n^3-5*n^4)/24
T(n+6,n)=(2400-2292*n-330*n^2+180*n^3+210*n^4+72*n^5)/120

with sketchy hints towards:
T(n+7,n)=(-16560+3600*n+17554*n^2+1395*n^3-3185*n^4-1755*n^5-329*n^6)/720

now it looks like the coefficients of the highest powers in n follow:
A000806 = 1,0,1,-5,36,-329,3655,-47844,721315,-12310199,234615096
aka
 Name:      Bessel polynomial y_n(-1).
 References :
 G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble
 fini totalement ordonne, Publications de l'Institut de Statistique
 de l'Universite de Paris, 23 (1978), 57-74.

Does this ring a bell with anyone?


Can anyone extend the 5-, 6- and 7-dimensional partitions beyond the current
A000390, A000416 and A000427 ?
that could definitively disprove this conjecture.
And disproving conjectures is, .., well, like *fun*

btw, the coefficients of the polynomials so far:
1
0, 1
2, 0, 0
0, 5, 0, 1
-48, 90, -7, -6, -5
1200, -1146, -165, 90, 105, 36
-16560, 3600, 17554, 1395, -3185, -1755, -329

not EIS-able yet, since the mere existence of the polynomials is
pure conjecture itself.

W.