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.

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A126460 Triangle T, read by rows, where column k of matrix power T^( k(k+1)/2 ) equals left-shifted column (k-1) of T for k>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 21, 21, 6, 1, 1, 274, 274, 75, 10, 1, 1, 5806, 5806, 1565, 195, 15, 1, 1, 182766, 182766, 48950, 5940, 420, 21, 1, 1, 8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1, 471517614, 471517614, 125727238, 14989472, 1006880
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1. Also, column k equals unsigned column k of the matrix inverse of triangle P_k defined by P_k(m,j) = C( C(j+2,3) - C(k+2,3) + m-j, m-j) for m>=j>=0.

Examples

			Triangle T begins:
1;
1, 1;
1, 1, 1;
3, 3, 1, 1;
21, 21, 6, 1, 1;
274, 274, 75, 10, 1, 1;
5806, 5806, 1565, 195, 15, 1, 1;
182766, 182766, 48950, 5940, 420, 21, 1, 1;
8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1; ...
where column 1 of T^1 equals left-shifted column 0 of T.
Matrix cube T^3 begins:
1;
3, 1;
6, 3, (1);
22, 12, (3), 1;
163, 91, (21), 3, 1;
2167, 1219, (274), 33, 3, 1;
46248, 26091, (5806), 661, 48, 3, 1;
1460301, 824853, (182766), 20341, 1369, 66, 3, 1; ...
where column 2 of T^3 equals left-shifted column 1 of T.
Matrix power T^6 begins:
1;
6, 1;
21, 6, 1;
98, 33, 6, (1);
791, 281, 51, (6), 1;
10850, 3929, 710, (75), 6, 1;
234472, 85557, 15425, (1565), 105, 6, 1;
7444172, 2725402, 490806, (48950), 3080, 141, 6, 1; ...
where column 3 of T^6 equals left-shifted column 2 of T.

Crossrefs

Columns: A126461, A126462, A126463, A126464; A126465 (dual); A107876 (variant); subpartitions defined: A115728.

Programs

  • PARI
    {T(n,k)=abs((matrix(n+1,n+1,r,c, binomial((c-1)*c*(c+1)/3!-k*(k+1)*(k+2)/3!+r-c,r-c))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
  • PARI
    /* As Defined by Matrix Product A126460 = A126445^-1*A126450: */
{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!,r-c))), N=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1,r-c)))); (M^-1*N)[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

G.f. of column k: 1/(1-x) = Sum_{n>=0} T(n+k,k)*x^n*(1-x)^p_k(n), so that column k equals the number of subpartitions of the partition p_k defined by: p_k(n) = (n^2 + (3*k+3)*n + (3*k^2+6*k-4))*n/6 for n>=0.

A126461 Column 0 of triangle A126460; equals the number of subpartitions of the partition {(k^2 + 3*k - 4)*k/6, k>=0}.

Original entry on oeis.org

1, 1, 1, 3, 21, 274, 5806, 182766, 8034916, 471517614, 35682799508, 3388864405941, 395127873991296, 55543575452873070, 9271180003481197642, 1813921568747948684475, 411378931233397975750296
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

When shifted left, equals column 1 of triangle A126460, which is the number of subpartitions of partition: {(k^2 + 6*k + 5)*k/6, k>=0}.

Examples

			Equals the number of subpartitions of the partition:
{(k^2 + 3*k - 4)*k/6, k>=0} = [0,0,2,7,16,30,50,77,112,156,210,275,...]
as illustrated by g.f.:
1/(1-x) = 1*(1-x)^0 + 1*x*(1-x)^0 + 1*x^2*(1-x)^2 + 3*x^3*(1-x)^7 + 21*x^4*(1-x)^16 + 274*x^5*(1-x)^30 + 5806*x^6*(1-x)^50 + 182766*x^7*(1-x)^77 ...

Crossrefs

Cf. A126460; A126462, A126463, A126464.

Programs

  • PARI
    {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(1+(k^2+3*k-4)*k/6)), n)}

Formula

G.f.: 1/(1-x) = Sum_{k>=0} a(k)*x^k*(1-x)^[(k^2 + 3*k - 4)*k/6].

A126463 Column 3 of triangle A126460; equals the number of subpartitions of the partition {(k^2 + 9*k + 20)*k/6, k>=0}.

Original entry on oeis.org

1, 1, 10, 195, 5940, 257300, 14989472, 1130000385, 107089958760, 12470885416545, 1751753684302150, 292264756622072214, 57165584968923450000, 12962148519535236156640, 3374220800446022166695530
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Examples

			Equals the number of subpartitions of the partition:
{(k^2 + 12*k + 41)*k/6, k>=0} = [0,9,23,43,70,105,149,203,268,345,...]
as illustrated by g.f.:
1/(1-x) = 1*(1-x)^0 + 1*x*(1-x)^9 + 10*x^2*(1-x)^23 + 195*x^3*(1-x)^43 + 5940*x^4*(1-x)^70 + 257300*x^5*(1-x)^105 + 14989472*x^6*(1-x)^149 + 1130000385*x^7*(1-x)^203 ...

Crossrefs

Cf. A126460; A126461, A126462, A126464.

Programs

  • PARI
    {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(1+(k^2+12*k+41)*k/6)),n)}

Formula

G.f.: 1/(1-x) = Sum_{k>=0} a(k)*x^k*(1-x)^[(k^2 + 12*k + 41)*k/6].

A126464 Row sums of triangle A126460.

Original entry on oeis.org

1, 2, 3, 8, 50, 635, 13389, 420865, 18491156, 1084804118, 82081329459, 7794746829520, 908790397019076, 127745867968533747, 21322592031518420776, 4171751138526111626665, 946103460280012610769060
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126460; A126461, A126462, A126463.

Programs

  • PARI
    {a(n)=sum(k=0,n,abs((matrix(n+1,n+1,r,c, binomial((c-1)*c*(c+1)/3!-k*(k+1)*(k+2)/3!+r-c,r-c))^-1)[n+1,k+1]))}
Showing 1-4 of 4 results.

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