cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A111810 Matrix log of triangle A098539, 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, 2, 2, 0, 10, 4, 4, 0, 88, 20, 8, 8, 0, 1096, 176, 40, 16, 16, 0, 11856, 2192, 352, 80, 32, 32, 0, -402480, 23712, 4384, 704, 160, 64, 64, 0, -1891968, -804960, 47424, 8768, 1408, 320, 128, 128, 0, 36024603264, -3783936, -1609920, 94848, 17536, 2816, 640, 256, 256, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 2^k times column 0 (A111811) when ignoring zeros above the diagonal.

Examples

			Matrix log of A098539, with factorial denominators, begins:
0;
1/1!, 0;
2/2!, 2/1!, 0;
10/3!, 4/2!, 4/1!, 0;
88/4!, 20/3!, 8/2!, 8/1!, 0;
1096/5!, 176/4!, 40/3!, 16/2!, 16/1!, 0;
11856/6!, 2192/5!, 352/4!, 80/3!, 32/2!, 32/1!, 0; ...

Crossrefs

Cf. A098539 (triangle), A111811 (column 0), A111813 (variant), A111941 (q=-1), A111843 (q=3), A111848 (q=4).

Programs

  • PARI
    T(n,k,q=2)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 2^k*T(n-k, 0) = 2^k*A111811(n-k) for n>=k>=0.

A111849 Column 0 of the matrix logarithm (A111848) of triangle A111845, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, 4, 56, 1728, -45696, -159401472, 387212983296, 14722642769657856, -783395638188945997824, -571756408840959817330851840, 603349161280921866200339538247680, 8390141848229920894318007084122311229440
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2005

Keywords

Comments

Let q=4; the g.f. of column k of A111845^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

Examples

			E.g.f. A(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4
- 45696/5!*x^5 - 159401472/6!*x^6 +...
where A(x) satisfies:
x = A(x) - A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3!
- A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
also:
Let G(x) be the g.f. of A111846 (column 0 of A111845), then
G(x) = 1 + x + 4*x^2 + 40*x^3 + 1040*x^4 + 78240*x^5 +...
= 1 + A(x) + A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3!
+ A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! +...

Crossrefs

Cf. A111848 (matrix log), A111845 (triangle), A111846, A111821 (variant), A111942 (q=-1), A111811 (q=2), A111844 (q=3).

Programs

  • PARI
    {a(n,q=4)=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]=(A^q)[i-1,1], 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]))}

Formula

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

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!.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Aug 25 2005

Keywords

Comments

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).

Examples

			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! +...

Crossrefs

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

Programs

  • PARI
    {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]))}

Formula

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).

A111812 Column 3 of triangle A098539, which shifts columns left and up under matrix square.

Original entry on oeis.org

1, 8, 72, 888, 16392, 479736, 23196168, 1909718520, 273790460424, 69532461669880, 31699923943776776, 26220200137673186808, 39689067731528646091272, 110732781183212424923225592
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2005

Keywords

Examples

			A(x) = 1 + 8*x + 72*x^2 + 888*x^3 + 16392*x^4 + 479736*x^5 +...

Crossrefs

Cf. A098539, A111811.

Programs

  • PARI
    {a(n,q=2)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+4,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]=(A^q)[i-1,1], B[i,j]=(A^q)[i-1,j-1]));));A=B);return(A[n+4,4]))}

Formula

G.f.: A(x) = 1 + Sum_{n>=1} 8^n/n!*Product_{j=0..n-1} L(2^j*x) where L(x) = e.g.f. of A111811 (column 0 of matrix log of A098539) satisfies: x = L(x) - L(x)*L(2*x)/2! + L(x)*L(2*x)*L(2^2*x)/3! - L(x)*L(2*x)*L(2^2*x)*L(2^3*x)/4! + ...
Showing 1-4 of 4 results.

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