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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A111825 Triangle P, read by rows, that satisfies [P^6](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(6*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(0,k)=1 and P(k,k)=1 for all k>=0. 8
1, 1, 1, 1, 6, 1, 1, 96, 36, 1, 1, 6306, 3816, 216, 1, 1, 1883076, 1625436, 139536, 1296, 1, 1, 2700393702, 3121837776, 360839016, 5036256, 7776, 1, 1, 19324893252552, 28794284803908, 4200503990976, 78293629296, 181382976, 46656, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also P(n,k) = the partitions of (6^n - 6^(n-k)) into powers of 6 <= 6^(n-k).

LINKS

Table of n, a(n) for n=0..35.

FORMULA

Let q=6; the g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} L(q^j*x) where L(x) satisfies: x/(1-x) = Sum_{n>=1} Product_{j=0..n-1} L(q^j*x)/(j+1) and L(x) equals the g.f. of column 0 of the matrix log of P (A111829).

EXAMPLE

Let q=6; the g.f. of column k of matrix power P^m is:

1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +

(m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +

(m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...

where L(x) satisfies:

x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...

and L(x) = x - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 +... (A111829).

Thus the g.f. of column 0 of matrix power P^m is:

1 + m*L(x) + m^2/2!*L(x)*L(6*x) + m^3/3!*L(x)*L(6*x)*L(6^2*x) +

m^4/4!*L(x)*L(6*x)*L(6^2*x)*L(6^3*x) + ...

Triangle P begins:

1;

1,1;

1,6,1;

1,96,36,1;

1,6306,3816,216,1;

1,1883076,1625436,139536,1296,1;

1,2700393702,3121837776,360839016,5036256,7776,1; ...

where P^6 shifts columns left and up one place:

1;

6,1;

96,36,1;

6306,3816,216,1; ...

PROG

(PARI) P(n, k, q=6)=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); return(A[n+1, k+1]))

CROSSREFS

Cf. A111826 (column 1), A111827 (row sums), A111828 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111830 (q=7), A111835 (q=8).

Sequence in context: A075377 A046792 A209330 * A085552 A002950 A324046

Adjacent sequences:  A111822 A111823 A111824 * A111826 A111827 A111828

KEYWORD

nonn,tabl

AUTHOR

Gottfried Helms and Paul D. Hanna, Aug 22 2005

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 September 20 06:51 EDT 2020. Contains 337264 sequences. (Running on oeis4.)