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.

A117165 Triangle of coefficients for the Shift-Moebius transform, read by rows.

Original entry on oeis.org

1, -1, 1, -2, 0, 1, -1, -1, 0, 1, -2, -1, 0, 0, 1, 1, -2, -1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 1, 3, -2, -1, -1, 0, 0, 0, 1, 0, 0, -2, -1, 0, 0, 0, 0, 1, 4, -2, -1, -1, -1, 0, 0, 0, 0, 1, 4, 0, -2, -1, -1, 0, 0, 0, 0, 0, 1, 5, 1, -1, -2, -1, -1, 0, 0, 0, 0, 0, 1, 1, 2, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 7, 0, 0, -2, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

Column k = Shift-Moebius transform of a sequence of all zeros except for a single '1' in position k: [0,0,0,..(k-1)zeros..,1,0,0,0,...].
Column 1 is A117166, the Shift-Moebius transform of [1,0,0,0,...].
Column 2 is A117167, the Shift-Moebius transform of [0,1,0,0,...].
Column 3 is A117168, the Shift-Moebius transform of [0,0,1,0,...].
Row sums give A117169, the Shift-Moebius transform of [1,1,1,...].

Examples

			Triangle begins:
1;
-1, 1;
-2, 0, 1;
-1,-1, 0, 1;
-2,-1, 0, 0, 1;
1,-2,-1, 0, 0, 1;
-1,-1,-1, 0, 0, 0, 1;
3,-2,-1,-1, 0, 0, 0, 1;
0, 0,-2,-1, 0, 0, 0, 0, 1;
4,-2,-1,-1,-1, 0, 0, 0, 0, 1;
4, 0,-2,-1,-1, 0, 0, 0, 0, 0, 1;
5, 1,-1,-2,-1,-1, 0, 0, 0, 0, 0, 1;
1, 2,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 1;
7, 0, 0,-2,-1,-1,-1, 0, 0, 0, 0, 0, 0, 1;
6, 3,-2,-1,-2,-1,-1, 0, 0, 0, 0, 0, 0, 0, 1;
5, 3, 1,-2,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 0, 1; ...

Crossrefs

Cf. A117166 (column 1), A117167 (column 2), A117168 (column 3), A117169 (row sums), A117170 (inverse), A117162, A008683; A117175.

Programs

  • PARI
    {T(n,k)=if(n=c,if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0))))[ n,k])}

Formula

The Shift-Moebius transform of a sequence B is equal to the limit of the iteration: let C_1 = B and for k>1, C_{k+1} = Moebius transform of C_k preceded by k zeros, then shift left k places (to drop the leading k zeros).
Triangle A117162 is a good example, starting with A008683 in column 1 as C_1 and each column k, C_k, is obtained using the above iteration, so that the columns converge to A117166.

A117166 Column 1 of triangle A117165 of Shift-Moebius coefficients and so equals the Shift-Moebius transform of [1,0,0,0,...].

Original entry on oeis.org

1, -1, -2, -1, -2, 1, -1, 3, 0, 4, 4, 5, 1, 7, 6, 5, 4, 6, 3, 5, 3, 2, 5, -3, -5, -1, 2, -7, -6, -13, -9, -14, -12, -22, -12, -25, -23, -26, -21, -35, -27, -38, -27, -43, -32, -47, -34, -51, -46, -52, -33, -53, -35, -58, -41, -56, -39, -54, -39, -61, -45, -53, -24, -46, -23, -44, -20, -41, -14, -32, -12, -22
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

Equals the self-convolution inverse of A117161, which is the limit of columns of triangle A112682.

Crossrefs

Cf. A117165 (triangle), A117167 (column 2), A117168 (column 3), A117169 (row sums), A117161 (inverse), A117160, A112682.

Programs

  • PARI
    {a(n)=prod(i=0,n, matrix(n,n,r,c,if(r>=c,if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0))))[ n,1]}

A117167 Column 2 of triangle A117165 of Shift-Moebius coefficients and so equals the Shift-Moebius transform of [0,1,0,0,...].

Original entry on oeis.org

0, 1, 0, -1, -1, -2, -1, -2, 0, -2, 0, 1, 2, 0, 3, 3, 4, 4, 6, 5, 7, 6, 7, 10, 7, 4, 7, 10, 7, 8, 5, 7, 6, 6, 1, 5, 0, -1, -2, -1, -8, -6, -11, -7, -15, -12, -20, -15, -24, -27, -33, -25, -36, -28, -41, -39, -48, -42, -51, -46, -61, -61, -69, -58, -72, -64, -77, -70, -85, -76, -90, -85, -94, -86, -99, -84, -100
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Crossrefs

Cf. A117165 (triangle), A117166 (column 1), A117168 (column 3), A117169 (row sums).

Programs

  • PARI
    {a(n)=prod(i=0,n,matrix(n+1,n+1,r,c,if(r>=c, if((r+n+1-i)%(c+n+1-i)==0,moebius((r+n+1-i)/(c+n+1-i)),0))))[n,2]}

A117168 Column 3 of triangle A117165 of Shift-Moebius coefficients and so equals the Shift-Moebius transform of [0,0,1,0,...].

Original entry on oeis.org

0, 0, 1, 0, 0, -1, -1, -1, -2, -1, -2, -1, -1, 0, -2, 1, 0, 2, 1, 4, 1, 5, 3, 6, 6, 7, 4, 8, 8, 10, 8, 10, 8, 12, 9, 10, 10, 10, 8, 10, 8, 9, 6, 7, 7, 5, 5, 3, 4, 1, -4, -5, -4, -7, -6, -13, -12, -17, -15, -20, -18, -28, -29, -37, -31, -40, -37, -47, -42, -55, -51, -62, -59, -69, -65, -78, -69, -84, -75, -93
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Crossrefs

Cf. A117165 (triangle), A117166 (column 1), A117167 (column 2), A117169 (row sums).

Programs

  • PARI
    {a(n)=prod(i=0,n+2,matrix(n+2,n+2,r,c,if(r>=c, if((r+n+2-i)%(c+n+2-i)==0,moebius((r+n+2-i)/(c+n+2-i)),0))))[n,3]}

A117160 Column 1 of triangle A112682; also equals row sums of A112682 (with offset).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 94, 210, 464, 1035, 2295, 5111, 11352, 25259, 56145, 124888, 277669, 617554, 1373201, 3053883, 6790995, 15102178, 33583784, 74684504, 166082706, 369337117, 821331578, 1826484804, 4061741926, 9032530513
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Feb 28 2006

Keywords

Comments

G.f.: A(x) = g.f. of A117169 (Shift-Moebius[1,1,1,1,...]) divided by the g.f. of A117166 (Shift-Moebius[1,0,0,0,...]) (see A117165 for the Shift-Moebius transform coefficients).
Limit_{n->oo} a(n+1)/a(n) = 2.223805416529545241557...

Crossrefs

Cf. A112682, A117165 (Shift-Moebius), A117169, A117166, A117161.

Programs

  • PARI
    {a(n)=if(n<1,0,SM=prod(i=0,n,matrix(n,n,r,c,if(r>=c, if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0)))); U=SM*vector(n,i,1)~;V=SM*vector(n,i,if(i==1,1,0))~; return(Vec(Ser(U)/Ser(V))[n]))}
Showing 1-5 of 5 results.

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