The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A111820 Triangle P, read by rows, that satisfies [P^5](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(5*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. 7
 1, 1, 1, 1, 5, 1, 1, 55, 25, 1, 1, 2055, 1525, 125, 1, 1, 291430, 311525, 38875, 625, 1, 1, 165397680, 239305275, 40338875, 975625, 3125, 1, 1, 390075741430, 735920617775, 157056792000, 5077475625, 24409375, 15625, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also P(n,k) = the partitions of (5^n - 5^(n-k)) into powers of 5 <= 5^(n-k). LINKS Table of n, a(n) for n=0..35. FORMULA Let q=5; 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 (A111824). EXAMPLE Let q=5; 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 - 3/2!*x^2 + 16/3!*x^3 + 2814/4!*x^4 +... (A111824). Thus the g.f. of column 0 of matrix power P^m is: 1 + m*L(x) + m^2/2!*L(x)*L(5*x) + m^3/3!*L(x)*L(5*x)*L(5^2*x) + m^4/4!*L(x)*L(5*x)*L(5^2*x)*L(5^3*x) + ... Triangle P begins: 1; 1,1; 1,5,1; 1,55,25,1; 1,2055,1525,125,1; 1,291430,311525,38875,625,1; 1,165397680,239305275,40338875,975625,3125,1; ... where P^5 shifts columns left and up one place: 1; 5,1; 55,25,1; 2055,1525,125,1; 291430,311525,38875,625,1; ... PROG (PARI) P(n, k, q=5)=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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)