A111975 -id:A111975 - 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).

A111976 Column 0 of triangle A111975, which shifts columns left and up under matrix square.

Original entry on oeis.org

1, 1, 1, 4, 16, 96, 896, 13568, 345088, 15112192, 1159913472, 158164664320, 38737429987328, 17197276791701504, 13946909814794223616, 20801835304287183306752, 57394078732651064041930752

Offset: 0

Paul D. Hanna, Aug 25 2005

nonn

			G.f. A(x) = 1 + x + x^2 + 4*x^3 + 16*x^4 + 96*x^5 + 896*x^6 +...
= 1 + L(x) + L(x)*L(2*x)/2! + L(x)*L(2*x)*L(2^2*x)/3! +...
where L(x) = x + 16/3!*x^3 + 1536/5!*x^5 - 319488/7!*x^7 +-...

Cf. A111975 (triangle), A111979.

{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);return(A[n+1,1]))}

G.f.: A(x) = 1 + Sum_{n>=1} (1/n!)*Product_{j=0..n-1} L(2^j*x) where L(x) satisfies: x-x^2 = Sum_{j>=1}(1-2^j*x)*Prod_{i=0..j-1}L(2^i*x); and L(x) equals the g.f. of column 0 of the matrix log of A111975 (A111979).

A111978 Matrix log of triangle A111975, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 16, 0, 4, 0, 0, 32, 0, 8, 0, 1536, 0, 64, 0, 16, 0, 0, 3072, 0, 128, 0, 32, 0, -319488, 0, 6144, 0, 256, 0, 64, 0, 0, -638976, 0, 12288, 0, 512, 0, 128, 0, 36007575552, 0, -1277952, 0, 24576, 0, 1024, 0, 256, 0, 0, 72015151104, 0, -2555904, 0, 49152, 0, 2048, 0, 512, 0

Offset: 0

Paul D. Hanna, Aug 25 2005

Column k equals 2^k multiplied by column 0 (A111979) when ignoring zeros above the diagonal.

			Matrix log of A111975, with factorial denominators, begins:
0;
1/1!, 0;
0/2!, 2/1!, 0;
16/3!, 0/2!, 4/1!, 0;
0/4!, 32/3!, 0/2!, 8/1!, 0;
1536/5!, 0/4!, 64/3!, 0/2!, 16/1!, 0;
0/6!, 3072/5!, 0/4!, 128/3!, 0/2!, 32/1!, 0;
-319488/7!, 0/6!, 6144/5!, 0/4!, 256/3!, 0/2!, 64/1!, 0; ...

Cf. A111975 (triangle), A111979 (column 0).

T(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); B=sum(i=1,#A,-(A^0-A)^i/i);return((n-k)!*B[n+1,k+1]))

T(n, k) = 2^k*T(n-k, 0) = 2^k*A111979(n-k) for n>=k>=0.

A111977 Row sums of triangle A111975, which shifts columns left and up under matrix square.

Original entry on oeis.org

1, 2, 4, 13, 57, 369, 3681, 59073, 1579393, 72188673, 5749089793, 809616264193, 204018868459521, 92907742733348865, 77097057406948106241, 117413997231333438701569, 330195264668839727287861249

Offset: 0

Paul D. Hanna, Aug 25 2005

nonn

Cf. A111975 (triangle), A078537 (variant).

{a(n,q=2)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,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); return(sum(k=0,n,A[n+1,k+1])))}

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!.

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A111976 Column 0 of triangle A111975, which shifts columns left and up under matrix square.

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A111978 Matrix log of triangle A111975, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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A111977 Row sums of triangle A111975, which shifts columns left and up under matrix square.

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