A111978 -id:A111978 - cp's OEIS frontend

A111979 Column 0 of the matrix logarithm (A111978) of triangle A111975, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

0, 1, 0, 16, 0, 1536, 0, -319488, 0, 36007575552, 0, -53682434054553600, 0, 1790644857560674043166720, 0, -1280831660558056667387645027942400, 0, 18961467116136182692294341450867551502336000, 0

Offset: 0

Paul D. Hanna, Aug 25 2005

sign

Let q=2; the g.f. of column k of A111975^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

			E.g.f. A(x) = x + 16/3!*x^3 + 1536/5!*x^5 - 319488/7!*x^7
+ 36007575552/9!*x^9 - 53682434054553600/11!*x^11 +...
where A(x) satisfies:
x*(1-x) = (1-2*x)*A(x) + (1-2^2*x)*A(x)*A(2*x)/2!
+ (1-2^3*x)*A(x)*A(2*x)*A(2^2*x)/3! +...

Cf. A111978 (matrix log), A111975 (triangle), A111976, A111811 (variant), A111814 (variant).

{a(n,q=2)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,j]=if(i>2,(A^q)[i-1,2],1), B[i,j]=(A^q)[i-1,j-1]));));A=B); B=sum(i=1,#A,-(A^0-A)^i/i);return(n!*B[n+1,1]))}

E.g.f. A(x): x-x^2 = Sum_{j>=1}(1-2^j*x)/j!*Prod_{i=0..j-1}A(2^i*x). E.g.f. A(x): x+x^2 = Sum_{j>=1}(1-4^j*x^2)/j!*Prod_{i=0..j-1}A(2^i*x).

A111975 Triangle P, read by rows, that satisfies [P^2](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(2*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(k,k)=1 and P(k+2,2)=P(k+2,0) for k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 4, 4, 1, 16, 16, 16, 8, 1, 96, 96, 96, 64, 16, 1, 896, 896, 896, 704, 256, 32, 1, 13568, 13568, 13568, 11776, 5504, 1024, 64, 1, 345088, 345088, 345088, 317952, 178176, 43776, 4096, 128, 1, 15112192, 15112192, 15112192, 14422016

Offset: 0

Paul D. Hanna, Aug 24 2005

Terms of column 0, column 1 and column 2 in row n are equal for n>2.

			Triangle P begins:
1;
1,1;
1,2,1;
4,4,4,1;
16,16,16,8,1;
96,96,96,64,16,1;
896,896,896,704,256,32,1;
13568,13568,13568,11776,5504,1024,64,1;
345088,345088,345088,317952,178176,43776,4096,128,1; ...
where P^2 shifts columns left and up one place:
1;
2,1;
4,4,1;
16,16,8,1;
96,96,64,16,1; ...
The matrix inverse, P^-1, equals signed P:
1;
-1,1;
1,-2,1;
-4,4,-4,1;
16,-16,16,-8,1; ...

Cf. A111976 (column 0), A111977 (row sums), A111978 (matrix log), A098539 (variant), A078536 (variant).

P(n,k,q=2)=local(A=Mat(1),B);if(n2,(A^q)[i-1,2],1), B[i,j]=(A^q)[i-1,j-1]));));A=B);return(A[n+1,k+1]))

The g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*2^k)^n/n! * Product_{j=0..n-1} L(2^j*x) where L(x) is the g.f. of column 0 of the matrix log of P (A111979) and satisfies: x-x^2 = Sum_{j>=1}(1-2^j*x)*Prod_{i=0..j-1}L(2^i*x).

cp's OEIS Frontend

A111979 Column 0 of the matrix logarithm (A111978) of triangle A111975, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula