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 1-8 of 8 results.

A114156 Triangle, read by rows, equal to the matrix inverse of P=A113370.

Original entry on oeis.org

1, -1, 1, 3, -4, 1, 6, 0, -7, 1, -8, 38, -21, -10, 1, -501, 692, -119, -60, -13, 1, -13623, 14910, -420, -735, -117, -16, 1, -409953, 401802, 22911, -12470, -2080, -192, -19, 1, -14544683, 13278520, 1577527, -255570, -51064, -4424, -285, -22, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Examples

			Triangle P^-1 begins:
1;
-1,1;
3,-4,1;
6,0,-7,1;
-8,38,-21,-10,1;
-501,692,-119,-60,-13,1;
-13623,14910,-420,-735,-117,-16,1;
-409953,401802,22911,-12470,-2080,-192,-19,1; ...
Triangle P^-2 begins:
1;
-2,1;
10,-8,1;
-9,28,-14,1;
-177,160,28,-20,1;
-2307,1366,455,10,-26,1;
-38874,15982,8666,660,-26,-32,1; ...

Crossrefs

Cf. A114157 (column 0), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); (P^-1)[n+1,k+1]

A114150 Triangle, read by rows, given by the product R^2*Q^-1 = Q^3*P^-2 using triangular matrices P=A113370, Q=A113381, R=A113389.

Original entry on oeis.org

1, 4, 1, 28, 7, 1, 326, 91, 10, 1, 5702, 1722, 190, 13, 1, 136724, 43764, 4945, 325, 16, 1, 4226334, 1415799, 163705, 10751, 496, 19, 1, 161385532, 56096733, 6617605, 437723, 19896, 703, 22, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114151, which gives R^-2*Q^3 = Q^-1*P^2.

Examples

			Triangle R^2*Q^-1 = Q^3*P^-2 begins:
1;
4,1;
28,7,1;
326,91,10,1;
5702,1722,190,13,1;
136724,43764,4945,325,16,1;
4226334,1415799,163705,10751,496,19,1; ...
Compare to P (A113370):
1;
1,1;
1,4,1;
1,28,7,1;
1,326,91,10,1;
1,5702,1722,190,13,1; ...
Thus R^2*Q^-1 = Q^3*P^-2 equals P shift left one column.

Crossrefs

Cf. A113370 (P), A113381 (Q), A113389 (R); A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (R^2*Q^-1)[n+1,k+1]

A114151 Triangle, read by rows, given by the product R^-2*Q^3 = Q^-1*P^2 using triangular matrices P=A113370, Q=A113381, R=A113389.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 15, 6, 1, 0, 136, 66, 9, 1, 0, 1998, 1091, 153, 12, 1, 0, 41973, 24891, 3621, 276, 15, 1, 0, 1166263, 737061, 110637, 8482, 435, 18, 1, 0, 40747561, 27110418, 4176549, 323874, 16430, 630, 21, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114150, which gives R^2*Q^-1 = Q^3*P^-2.

Examples

			Triangle R^-2*Q^3 = Q^-1*P^2 begins:
1;
0,1;
0,3,1;
0,15,6,1;
0,136,66,9,1;
0,1998,1091,153,12,1;
0,41973,24891,3621,276,15,1; ...
Compare to R (A113389):
1;
3,1;
15,6,1;
136,66,9,1;
1998,1091,153,12,1;
41973,24891,3621,276,15,1; ...
Thus R^-2*Q^3 = Q^-1*P^2 equals R shift right one column.

Crossrefs

Cf. A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (Q^-1*P^2)[n+1,k+1]

A114152 Triangle, read by rows, given by the product R^3*P^-1 using triangular matrices P=A113370, R=A113389.

Original entry on oeis.org

1, 8, 1, 84, 14, 1, 1296, 252, 20, 1, 27850, 5957, 510, 26, 1, 784146, 179270, 16180, 858, 32, 1, 27630378, 6641502, 623115, 34125, 1296, 38, 1, 1177691946, 294524076, 28470525, 1599091, 61952, 1824, 44, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114153, which gives R^-1*P^3.

Examples

			Triangular matrix R^3*P^-1 begins:
1;
8,1;
84,14,1;
1296,252,20,1;
27850,5957,510,26,1;
784146,179270,16180,858,32,1;
27630378,6641502,623115,34125,1296,38,1; ...
Compare to P^2 (A113374):
1;
2,1;
6,8,1;
37,84,14,1;
429,1296,252,20,1;
7629,27850,5957,510,26,1; ...
Thus R^3*P^-1 equals P^2 shift left one column.

Crossrefs

Cf. A113374 (P^2), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (R^3*P^-1)[n+1,k+1]

A114153 Triangle, read by rows, given by the product R^-1*P^3 using triangular matrices P=A113370, R=A113389.

Original entry on oeis.org

1, 0, 1, 0, 6, 1, 0, 48, 12, 1, 0, 605, 186, 18, 1, 0, 11196, 3892, 414, 24, 1, 0, 280440, 106089, 12021, 732, 30, 1, 0, 8981460, 3620379, 429345, 27152, 1140, 36, 1, 0, 353283128, 149740555, 18386361, 1196910, 51445, 1638, 42, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114152, which gives R^3*P^-1.

Examples

			Triangle R^-1*P^3 begins:
1;
0,1;
0,6,1;
0,48,12,1;
0,605,186,18,1;
0,11196,3892,414,24,1;
0,280440,106089,12021,732,30,1; ...
Compare to R^2 (A113392):
1;
6,1;
48,12,1;
605,186,18,1;
11196,3892,414,24,1;
280440,106089,12021,732,30,1; ...
Thus R^-1*P^3 equals R^2 shift right one column.

Crossrefs

Cf. A113392 (R^2), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (R^-1*P^3)[n+1,k+1]

A114154 Triangle, read by rows, given by the product R^3*Q^-2 using triangular matrices Q=A113381, R=A113389.

Original entry on oeis.org

1, 5, 1, 45, 8, 1, 635, 120, 11, 1, 12815, 2556, 231, 14, 1, 343815, 71548, 6556, 378, 17, 1, 11651427, 2508528, 233706, 13391, 561, 20, 1, 480718723, 106427700, 10069521, 579047, 23817, 780, 23, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114155, which gives Q^-2*P^3.

Examples

			Triangle R^3*Q^-2 begins:
1;
5,1;
45,8,1;
635,120,11,1;
12815,2556,231,14,1;
343815,71548,6556,378,17,1; ...
Compare to Q (A113381):
1;
2,1;
6,5,1;
37,45,8,1;
429,635,120,11,1;
7629,12815,2556,231,14,1; ...
Thus R^3*Q^-2 equals Q shift left one column.

Crossrefs

Cf. A113394 (R^3), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (R^3*Q^-2)[n+1,k+1]

A114155 Triangle, read by rows, given by the product Q^-2*P^3 using triangular matrices P=A113370, Q=A113381.

Original entry on oeis.org

1, -1, 1, 3, 2, 1, 6, 6, 5, 1, -8, 37, 45, 8, 1, -501, 429, 635, 120, 11, 1, -13623, 7629, 12815, 2556, 231, 14, 1, -409953, 185776, 343815, 71548, 6556, 378, 17, 1, -14544683, 5817106, 11651427, 2508528, 233706, 13391, 561, 20, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114154, which gives R^3*Q^-2. Column 0 equals column 0 of P^-1 (A114157).

Examples

			Triangle Q^-2*P^3 begins:
1;
-1,1;
3,2,1;
6,6,5,1;
-8,37,45,8,1;
-501,429,635,120,11,1;
-13623,7629,12815,2556,231,14,1;
-409953,185776,343815,71548,6556,378,17,1; ...
Compare to Q (A113381):
1;
2,1;
6,5,1;
37,45,8,1;
429,635,120,11,1;
7629,12815,2556,231,14,1;...
Thus Q^-2*P^3 shift left one column equals Q.

Crossrefs

Cf. A114157 (column 0), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (Q^-2*P^3)[n+1,k+1]

A114158 Triangle, read by rows, equal to the matrix inverse of Q=A113381.

Original entry on oeis.org

1, -2, 1, 4, -5, 1, 21, -5, -8, 1, 130, 20, -32, -11, 1, 1106, 840, -260, -77, -14, 1, 10044, 24865, -2584, -1089, -140, -17, 1, -18366, 823383, -12828, -21428, -2737, -221, -20, 1, -9321125, 31847653, 1160956, -523831, -73458, -5474, -320, -23, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Examples

			Triangle Q^-1 begins:
1;
-2,1;
4,-5,1;
21,-5,-8,1;
130,20,-32,-11,1;
1106,840,-260,-77,-14,1;
10044,24865,-2584,-1089,-140,-17,1;
-18366,823383,-12828,-21428,-2737,-221,-20,1; ...
Triangle Q^-2 begins:
1;
-4,1;
18,-10,1;
20,30,-16,1;
-139,255,24,-22,1;
-3945,3085,544,0,-28,1;
-99849,51015,12444,671,-42,-34,1; ...

Crossrefs

Cf. A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114156 (P^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); (Q^-1)[n+1,k+1]
Showing 1-8 of 8 results.

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