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 A118190 Triangle T, read by rows, defined by: T(n,k) = (5^k)^(n-k) for n>=k>=0. 12
 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; text; internal format)
 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). LINKS 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 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). PROG (PARI) T(n, k)=if(n

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Last modified January 17 09:54 EST 2020. Contains 330949 sequences. (Running on oeis4.)