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

 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 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

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.

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