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A118190
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Triangle T, read by rows, defined by: T(n,k) = (5^k)^(n-k) for n>=k>=0. 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).
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11
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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).
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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).
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EXAMPLE
| A(x,y) = 1/(1-xy) + x/(1-5xy) + x^2/(1-25xy) + x^3/(1-125xy) + ...
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).
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PROG
| (PARI) T(n, k)=if(n<k|k<0, 0, (5^k)^(n-k) )
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CROSSREFS
| Cf. A118191 (row sums), A118192 (antidiagonal sums); A118193, A118194; variants: A117401 (q=2), A118180 (q=3), A118185 (q=4).
Sequence in context: A152572 A203346 A176793 * A172342 A143213 A172377
Adjacent sequences: A118187 A118188 A118189 * A118191 A118192 A118193
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2006
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