A152285 Triangle T, read by rows, where g.f. of row n of matrix power T^n = (n^3 + y)*y^(n-1) for n>=0.

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

Offset

0,4

Links

Table of n, a(n) for n=0..36.

Formula

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.

Example

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
...

Prog

(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]}

Crossrefs

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

Sequence in context: A030043 A045497 A117258 * A009417 A009451 A321330

Adjacent sequences: A152282 A152283 A152284 * A152286 A152287 A152288

Keyword

sign,tabl

Author

Paul D. Hanna, Dec 09 2008

Status

approved