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A118190
Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.
20
1, 1, 1, 1, 5, 1, 1, 25, 25, 1, 1, 125, 625, 125, 1, 1, 625, 15625, 15625, 625, 1, 1, 3125, 390625, 1953125, 390625, 3125, 1, 1, 15625, 9765625, 244140625, 244140625, 9765625, 15625, 1, 1, 78125, 244140625, 30517578125, 152587890625, 30517578125, 244140625, 78125, 1
OFFSET
0,5
COMMENTS
Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character. For example, the matrix inverse is defined by [T^-1](n,k) = A118193(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118194(n-k)*T(n,k).
For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-5^n*x).
FORMULA
G.f.: A(x,y) = Sum_{n>=0} x^n/(1-5^n*x*y).
G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,5*y).
T(n,k) = (1/n)*( 5^(n-k)*k*T(n-1,k-1) + 5^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 21 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 3. - G. C. Greubel, Jun 29 2021
EXAMPLE
A(x,y) = 1/(1-x*y) + x/(1-5*x*y) + x^2/(1-25*x*y) + x^3/(1-125*x*y) + ...
Triangle begins:
1;
1, 1;
1, 5, 1;
1, 25, 25, 1;
1, 125, 625, 125, 1;
1, 625, 15625, 15625, 625, 1;
1, 3125, 390625, 1953125, 390625, 3125, 1;
1, 15625, 9765625, 244140625, 244140625, 9765625, 15625, 1; ...
The matrix inverse T^-1 starts:
1;
-1, 1;
4, -5, 1;
-76, 100, -25, 1;
7124, -9500, 2500, -125, 1;
-3326876, 4452500, -1187500, 62500, -625, 1; ...
where [T^-1](n,k) = A118193(n-k)*(5^k)^(n-k).
MATHEMATICA
With[{m=3}, Table[(m+2)^(k*(n-k)), {n, 0, 12}, {k, 0, n}]//Flatten] (* G. C. Greubel, Jun 29 2021 *)
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, (5^k)^(n-k) )
(Magma) [5^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 29 2021
(Sage) flatten([[5^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 29 2021
CROSSREFS
Cf. A118191 (row sums), A118192 (antidiagonal sums), A118193, A118194.
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), this sequence (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).
Sequence in context: A152572 A203346 A176793 * A172342 A143213 A172377
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 15 2006
STATUS
approved