A111830 Triangle P, read by rows, that satisfies [P^7](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(7*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.

1, 1, 1, 1, 7, 1, 1, 154, 49, 1, 1, 16275, 8281, 343, 1, 1, 9106461, 6558209, 410914, 2401, 1, 1, 28543862991, 27307109501, 2298650515, 20170801, 16807, 1, 1, 521136519414483, 636922972420469, 67522139062441, 790856748801, 988621354

Offset

0,5

Comments

Also P(n,k) = partitions of (7^n - 7^(n-k)) into powers of 7 <= 7^(n-k).

Links

Table of n, a(n) for n=0..33.

Formula

Let q=7; 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 (A111834).

Example

Let q=7; 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 - 5/2!*x^2 + 83/3!*x^3 + 16110/4!*x^4 +... (A111834).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(7*x) + m^3/3!*L(x)*L(7*x)*L(7^2*x) +
m^4/4!*L(x)*L(7*x)*L(7^2*x)*L(7^3*x) + ...
Triangle P begins:
1;
1,1;
1,7,1;
1,154,49,1;
1,16275,8281,343,1;
1,9106461,6558209,410914,2401,1;
1,28543862991,27307109501,2298650515,20170801,16807,1; ...
where P^7 shifts columns left and up one place:
1;
7,1;
154,49,1;
16275,8281,343,1; ...

Prog

(PARI) P(n, k, q=7)=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. A111831 (column 1), A111832 (row sums), A111833 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111825 (q=6), A111835 (q=8).

Sequence in context: A156692 A188644 A331899 * A212943 A174588 A366316

Adjacent sequences: A111827 A111828 A111829 * A111831 A111832 A111833

Keyword

frac,nonn,tabl

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Status

approved