login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=0..41.

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<k || k<0, 0, (5^k)^(n-k) )

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

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 15 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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