editing

approved

(PARI) {T(n, k, q=5)=local(A=Mat(1), B); if(n<k || k<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return((n-k)!*B[n+1, k+1]))}

approved

editing

editing

approved

Matrix log of triangle A111820, which shifts columns left and up under matrix 5-th 5th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

approved

editing

Gottfried Helms and _Paul D. Hanna (pauldhanna(AT)juno.com), _, Aug 22 2005

_Gottfried Helms (helms(AT)uni-kassel.de) _ and Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2005

frac,sign,tabl,new

Gottfried Helms (helms(AT)uni-kassel.de) and Paul D . Hanna (pauldhanna(AT)juno.com), Aug 22 2005

T(n, k) = 5^k*T(n-k, 0) = A111824(n-k) for n>=k>=0.

frac,sign,tabl,new

0,54

(PARI) {T(n, k, q=5)=local(A=Mat(1), B); if(n<k|k<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i|j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return((n-k)!*B[n+1, k+1]))}

frac,sign,tabl,new

Matrix log of triangle A111820, which shifts columns left and up under matrix 5-th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -3, 5, 0, 16, -15, 25, 0, 2814, 80, -75, 125, 0, -1092180, 14070, 400, -375, 625, 0, -3603928080, -5460900, 70350, 2000, -1875, 3125, 0, 58978973128440, -18019640400, -27304500, 351750, 10000, -9375, 15625, 0, 5974833380453777520

0,5

Column k equals 5^k multiplied by column 0 (A111824) when ignoring zeros above the diagonal.

T(n,k) = 5^k*T(n-k,0) = A111824(n-k) for n>=k>=0.

Matrix log of A111820, with factorial denominators, begins:

0;

1/1!, 0;

-3/2!, 5/1!, 0;

16/3!, -15/2!, 25/1!, 0;

2814/4!, 80/3!, -75/2!, 125/1!, 0;

-1092180/5!, 14070/4!, 400/3!, -375/2!, 625/1!, 0; ...

(PARI) {T(n, k, q=5)=local(A=Mat(1), B); if(n<k|k<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i|j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return((n-k)!*B[n+1, k+1]))}

Cf. A111820, A111824 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111828 (q=6), A111833 (q=7), A111838 (q=8).

frac,sign,tabl

Gottfried Helms (helms(AT)uni-kassel.de) and Paul D Hanna (pauldhanna(AT)juno.com), Aug 22 2005

approved