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 1-5 of 5 results.

A102086 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (n+1).

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 16, 20, 9, 4, 127, 156, 63, 16, 5, 1363, 1664, 648, 144, 25, 6, 18628, 22684, 8703, 1840, 275, 36, 7, 311250, 378572, 144243, 29824, 4200, 468, 49, 8, 6173791, 7504640, 2849400, 582640, 79775, 8316, 735, 64, 9, 142190703, 172785512
Offset: 0

Views

Author

Paul D. Hanna, Dec 29 2004

Keywords

Comments

Column 0 forms A082161. Column 1 forms A102087. Row sums form A102088.

Examples

			Rows of T begin:
[1],
[1,2],
[3,4,3],
[16,20,9,4],
[127,156,63,16,5],
[1363,1664,648,144,25,6],
[18628,22684,8703,1840,275,36,7],
[311250,378572,144243,29824,4200,468,49,8],
[6173791,7504640,2849400,582640,79775,8316,735,64,9],...
Matrix square T^2 equals T excluding the main diagonal:
[1],
[3,4],
[16,20,9],
[127,156,63,16],
[1363,1664,648,144,25],...
G.f. for column 0: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-3x) + ... + T(n,0)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ...
G.f. for column 1: 2 = 2(1-2x) + 4*x*(1-2x)(1-3x) + 20*x^2*(1-2x)(1-3x)(1-4x) + ... + T(n+1,1)*x^n*(1-2x)(1-3x)(1-4x)*..*(1-(n+2)*x) + ...
G.f. for column 2: 3 = 3(1-3x) + 9*x*(1-3x)(1-4x) + 63*x^2*(1-3x)(1-4x)(1-5x) + ... + T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ...

Crossrefs

Cf. A082161, A102087, A102088.
Cf. A102316.

Programs

  • Maple
    {T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,k+1])}
  • Mathematica
    T[n_, n_] := n+1; T[n_, k_] /; k>n = 0; T[n_, k_] /; k == n-1 := n^2; T[n_, k_] := T[n, k] = Coefficient[1-Sum[T[i, k]*x^i*Product[1-(j+k)*x, {j, 1, i-k+1}], {i, k, n-1}], x, n]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 15 2014, after PARI script *)
  • PARI
    {T(n,k)=if(n

Formula

T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1.
G.f. for column k: T(k, k) = k+1 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).

A102088 Row sums of triangular matrix A102086, which shifts upward to exclude the main diagonal under matrix square.

Original entry on oeis.org

1, 3, 10, 49, 367, 3850, 52173, 868614, 17199370, 395757887, 10396896795, 307511681367, 10124396776169, 367567456615441, 14598938725992903, 630060602243145513, 29375322688053255480, 1472008290120323375502
Offset: 0

Views

Author

Paul D. Hanna, Dec 29 2004

Keywords

Crossrefs

Cf. A102086, A102087.

Programs

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

A102321 Column 0 of triangular matrix A102320, which satisfies T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).

Original entry on oeis.org

1, 1, 4, 33, 436, 8122, 197920, 6007205, 219413116, 9402081718, 463548752912, 25893783163498, 1618536618626888, 112053082721454708, 8518619080226661504, 705977323976245345133, 63382036275445226941548
Offset: 0

Views

Author

Paul D. Hanna, Jan 05 2005

Keywords

Examples

			G.f.: 1 = (1-x) + 1*x*(1-x)(1-3x) + 4*x^2*(1-x)(1-3x)(1-5x) + ... + a(n)*x^n*(1-x)(1-3x)(1-5x)*..*(1-(2n+1)*x) + ...

Crossrefs

Cf. A102087, A102323.

Programs

  • PARI
    {a(n)=local(A=Mat(1),B); for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=2*j-1,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,1])}
  • PARI
    {a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=0,k,1-(2*j+1)*x+x*O(x^n))),n))}

Formula

G.f.: 1 = Sum_{n>=0} a(n)*x^n*prod_{k=0, n} (1-(2k+1)*x) for n>0 with a(0)=1.

A102322 Column 1 of triangular matrix A102320, which that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).

Original entry on oeis.org

0, 3, 9, 72, 945, 17568, 427770, 12979080, 473981445, 20308813128, 1001231706582, 55927084380552, 3495759750651978, 242012640619081152, 18398411206663695732, 1524754064472700613328, 136890662566189661556525
Offset: 0

Views

Author

Paul D. Hanna, Jan 05 2005

Keywords

Crossrefs

Cf. A102087, A102320.

Programs

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

A106208 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 16, 10, 3, 1, 127, 78, 21, 4, 1, 1363, 832, 216, 36, 5, 1, 18628, 11342, 2901, 460, 55, 6, 1, 311250, 189286, 48081, 7456, 840, 78, 7, 1, 6173791, 3752320, 949800, 145660, 15955, 1386, 105, 8, 1, 142190703, 86392756, 21826470, 3327340
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Comments

Column 0 is A082161 (offset 1). Column 1 is (1/2)*A102087. Row sums form A106209.

Examples

			Triangle T begins:
1;
1,1;
3,2,1;
16,10,3,1;
127,78,21,4,1;
1363,832,216,36,5,1;
18628,11342,2901,460,55,6,1;
311250,189286,48081,7456,840,78,7,1;
6173791,3752320,949800,145660,15955,1386,105,8,1; ...
Matrix inverse T^-1 begins:
1;
-1,1;
-1,-2,1;
-3,-4,-3,1;
-16,-20,-9,-4,1;
-127,-156,-63,-16,-5,1;
-1363,-1664,-648,-144,-25,-6,1;
-18628,-22684,-8703,-1840,-275,-36,-7,1; ...
where [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0.
G.f. for column 0: 1 = 1(1-x) + 1*x*(1-x)(1-2x) +
3*x^2*(1-x)(1-2x)(1-3x) + ... +
T(n,0)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ...
G.f. for column 1: 1 = 1(1-2x) + 2*x*(1-2x)(1-3x) +
10*x^2*(1-2x)(1-3x)(1-4x) + ... +
T(n+1,1)*x^n*(1-2x)(1-3x)(1-4x)*..*(1-(n+2)*x) + ...
G.f. for column 2: 1 = 1(1-3x) + 3*x*(1-3x)(1-4x) +
21*x^2*(1-3x)(1-4x)(1-5x) + ... +
T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ...

Crossrefs

Cf. A102086, A082161, A106209.

Programs

  • PARI
    T(n,k)=if(n
  • PARI
    T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,k+1]/(k+1))

Formula

T(n, k) = A102086(n, k)/(k+1) for n>=0, k>=0. T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1. G.f. for column k: 1 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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