0,4

T(n,k) is divisible by C(n,k)^2 for n>=k>=0.

For matrix powers: [T^m](n,k)/C(n,k)^2 is an integer for all integer m and n>=k>=0.

Triangle T begins:

1;

1,1;

-2,4,1;

42,-36,9,1;

-2448,1584,-216,16,1;

297120,-165600,18000,-800,25,1;

-64276200,32619600,-3132000,114000,-2250,36,1;

22435915320,-10679785200,947268000,-30870000,507150,-5292,49,1;

-11785471407360,5358907814400,-449832700800,13651948800,-202507200,1778112,-10976,64,1;

8847028761338880,-3890309297817600,313670692339200,-9056483251200,125763321600,-1002855168,5249664,-20736,81,1;

...

Illustrate: g.f. of row n of T^n = (n^3 + y)*y^(n-1) as follows.

Matrix cube T^3 begins:

1;

3,1;

6,12,1;

0,0,27,1; <-- row 3 of T^3, g.f.: (3^3 + y)*y^2

...

Matrix power T^4 begins:

1;

4,1;

16,16,1;

-12,72,36,1;

0,0,0,64,1; <-- row 4 of T^4, g.f.: (4^3 + y)*y^3

...

Matrix power T^5 begins:

1;

5,1;

30,20,1;

30,180,45,1;

240,-720,360,80,1;

0,0,0,0,125,1; <-- row 5 of T^5, g.f.: (5^3 + y)*y^4

...

Matrix power T^6 begins:

1;

6,1;

48,24,1;

162,324,54,1;

-288,-576,864,96,1;

-8880,32400,-9000,1200,150,1;

0,0,0,0,0,216,1; <-- row 6 of T^6, g.f.: (6^3 + y)*y^5

...

(PARI) {T(n, k)=local(M, N); M=if(n==0, Mat(1), M=matrix(n, n, r, c, if(r>=c, T(r-1, c-1)))); for(j=1, n, N=matrix(#M+1, #M+1, r, c, if(r==c, 1, if(r>c, if(r<=#M, M[r, c], if(c==#M, c^2, 0))))); for(i=2, #N-1, N[ #N, #N-i]=-(N^(#N-1))[ #N, #N-i]/(#N-1)); M=N); M[n+1, k+1]}

Cf. columns: A152286, A152287, A152288; A152289; variant: A132870.

Sequence in context: A030043 A045497 A117258 this_sequence A009417 A009451 A077748

Adjacent sequences: A152282 A152283 A152284 this_sequence A152286 A152287 A152288

sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 09 2008