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 1-10 of 17 results. Next

A105623 Matrix square-root of triangle A105615.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 26, 10, 3, 1, 226, 74, 19, 4, 1, 2426, 706, 167, 31, 5, 1, 30826, 8162, 1831, 320, 46, 6, 1, 451586, 110410, 23843, 4021, 548, 64, 7, 1, 7489426, 1708394, 358339, 59024, 7801, 866, 85, 8, 1, 138722426, 29752066, 6097607, 987763, 127985
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

Column 0 equals A105616 (=column 1 of A105615) shift 1 place right. Column 1 is A000698 (related to double factorials) offset 1.

Examples

			Triangle begins:
1;
1,1;
4,2,1;
26,10,3,1;
226,74,19,4,1;
2426,706,167,31,5,1;
30826,8162,1831,320,46,6,1;
451586,110410,23843,4021,548,64,7,1;
7489426,1708394,358339,59024,7801,866,85,8,1;
138722426,29752066,6097607,987763,127985,13801,1289,109,9,1; ...

Crossrefs

Cf. A105615, A105616 (column 0), A000698 (column 1), A105620 (matrix inverse).

Programs

  • PARI
    T(n,k)=local(R,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-1); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n

A105620 Matrix inverse square-root of triangle A105615.

Original entry on oeis.org

1, -1, 1, -2, -2, 1, -10, -4, -3, 1, -74, -20, -7, -4, 1, -706, -148, -39, -11, -5, 1, -8162, -1412, -315, -70, -16, -6, 1, -110410, -16324, -3243, -635, -116, -22, -7, 1, -1708394, -220820, -40167, -7264, -1183, -180, -29, -8, 1, -29752066, -3416788, -579159, -99191, -15065, -2049, -265, -37, -9, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

Column 0 is negative A000698 (related to double factorials). Column 1 equals twice column 0 after the initial term.

Examples

			Triangle begins:
1;
-1,1;
-2,-2,1;
-10,-4,-3,1;
-74,-20,-7,-4,1;
-706,-148,-39,-11,-5,1;
-8162,-1412,-315,-70,-16,-6,1;
-110410,-16324,-3243,-635,-116,-22,-7,1;
-1708394,-220820,-40167,-7264,-1183,-180,-29,-8,1;
-29752066,-3416788,-579159,-99191,-15065,-2049,-265,-37,-9,1; ...

Crossrefs

Cf. A105615, A105619 (matrix square), A105623 (matrix inverse), A000698 (column 0), A105621 (column 2), A105622 (row sums).

Programs

  • PARI
    T(n,k)=local(R,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n

A105616 Column 1 of triangle A105615.

Original entry on oeis.org

1, 4, 26, 226, 2426, 30826, 451586, 7489426, 138722426, 2839238026, 63654973826, 1551919194226, 40888965122426, 1157981114051626, 35083865696279426, 1132449247218851026, 38800104353355372026, 1406432065083818193226
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

Also equals column 0 of triangle A105623 (offset 1), where A105623 equals the matrix square-root of triangle A105615.

Crossrefs

Cf. A105615, A105623.

Programs

  • PARI
    {a(n)=if(n<0,0,(matrix(n+2,n+2,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j) ))))^-1)[n+2,2])}

A105617 Column 2 of triangle A105615.

Original entry on oeis.org

1, 6, 50, 522, 6498, 93666, 1532970, 28079082, 569212578, 12655466946, 306280630890, 8017054975242, 225716319717858, 6802519195684386, 218521006115328810, 7454198349649868202, 269114811307118424738
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Crossrefs

Cf. A105615.

Programs

  • PARI
    {a(n)=if(n<0,0,(matrix(n+3,n+3,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j) ))))^-1)[n+3,3])}

A105619 Matrix inverse of triangle A105615.

Original entry on oeis.org

1, -2, 1, -2, -4, 1, -10, -2, -6, 1, -74, -10, -2, -8, 1, -706, -74, -10, -2, -10, 1, -8162, -706, -74, -10, -2, -12, 1, -110410, -8162, -706, -74, -10, -2, -14, 1, -1708394, -110410, -8162, -706, -74, -10, -2, -16, 1, -29752066, -1708394, -110410, -8162, -706, -74, -10, -2, -18, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

Except for the initial few terms, all columns are equal to negative A000698 (related to double factorials).

Examples

			Triangle begins:
1;
-2,1;
-2,-4,1;
-10,-2,-6,1;
-74,-10,-2,-8,1;
-706,-74,-10,-2,-10,1;
-8162,-706,-74,-10,-2,-12,1;
-110410,-8162,-706,-74,-10,-2,-14,1;
-1708394,-110410,-8162,-706,-74,-10,-2,-16,1; ...

Crossrefs

Cf. A105615, A000698, A105620 (matrix square-root).

Programs

  • PARI
    T(n,k)=if(n=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))[n+1,k+1])

A105626 Triangular matrix T, read by rows, that satisfies T^2 = A105615^3; also equals the matrix cube of triangle A105623.

Original entry on oeis.org

1, 3, 1, 18, 6, 1, 150, 48, 9, 1, 1566, 480, 93, 12, 1, 19494, 5736, 1125, 153, 15, 1, 280998, 79584, 15681, 2190, 228, 18, 1, 4598910, 1256808, 247929, 35181, 3780, 318, 21, 1, 84237246, 22262640, 4389213, 629424, 68961, 6000, 423, 24, 1, 1707637734
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

SHIFT_LEFT(column 0 of T) = 3*(column 2 of A105615). A105623 equals the matrix square-root of triangle A105615.

Examples

			Triangle begins:
1;
3,1;
18,6,1;
150,48,9,1;
1566,480,93,12,1;
19494,5736,1125,153,15,1;
280998,79584,15681,2190,228,18,1;
4598910,1256808,247929,35181,3780,318,21,1;
84237246,22262640,4389213,629424,68961,6000,423,24,1; ...

Crossrefs

Cf. A105615, A105617, A105623, A105627 (column 1), A105628 (row sums).

Programs

  • PARI
    T(n,k)=local(R,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-3); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n

Formula

T(n+1, 0) = 3*A105615(n+2, 2) = 3*A105617(n) for n>=0.

A105618 Row sums of triangle A105615.

Original entry on oeis.org

1, 3, 15, 107, 991, 11203, 148879, 2270027, 39041151, 747704963, 15784630159, 364256650027, 9124264794271, 246600188525123, 7153677209063439, 221729176945261067, 7313478624348963391, 255786689421734222083
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Crossrefs

Cf. A105615.

Programs

  • PARI
    {a(n)=if(n<0,0,sum(k=0,n,(matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-1)[n+1,k+1]))}

A107717 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((3*p-1)/3)) = (3*p-1)*(column p of T), or [T^((3*p-1)/3)](m,0) = (3*p-1)*T(p+m,p) for all m>=1 and p>=0.

Original entry on oeis.org

1, 3, 1, 21, 6, 1, 219, 57, 9, 1, 2973, 723, 111, 12, 1, 49323, 11361, 1713, 183, 15, 1, 964173, 212151, 31575, 3351, 273, 18, 1, 21680571, 4584081, 675489, 71391, 5799, 381, 21, 1, 551173053, 112480887, 16442823, 1732881, 140529, 9219, 507, 24, 1
Offset: 0

Views

Author

Paul D. Hanna, May 30 2005

Keywords

Comments

Column 0 is A107716 (INVERTi of triple factorials). Column 1 is A107718 (twcie column 0 of T^(2/3), offset 1). The matrix logarithm divided by 3 yields the integer triangle A107724.

Examples

			SHIFT_LEFT(column 0 of T^(p-1/3)) = (3*p-1)*(column p of T):
SHIFT_LEFT(column 0 of T^(-1/3)) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^(2/3)) = 2*(column 1 of T);
SHIFT_LEFT(column 0 of T^(5/3)) = 5*(column 2 of T).
Triangle begins:
1;
3,1;
21,6,1;
219,57,9,1;
2973,723,111,12,1;
49323,11361,1713,183,15,1;
964173,212151,31575,3351,273,18,1;
21680571,4584081,675489,71391,5799,381,21,1; ...
Matrix power (2/3), T^(2/3), is A107719 and begins:
1;
2,1;
12,4,1;
114,32,6,1;
1446,364,62,8,1;
22722,5276,854,102,10,1; ...
compare column 0 of T^(2/3) to 2*(column 1 of T).
Matrix inverse cube-root T^(-1/3) is A107727 and begins:
1;
-1,1;
-3,-2,1;
-21,-7,-3,1;
-219,-53,-13,-4,1;
-2973,-583,-115,-21,-5,1; ...
compare column 0 of T^(-1/3) to column 0 of T.
Matrix inverse is A107726 and begins:
1;
-3,1;
-3,-6,1;
-21,-3,-9,1;
-219,-21,-3,-12,1;
-2973,-219,-21,-3,-15,1; ...
compare column 0 of T^(-1) to column 0 of T.

Crossrefs

Cf. A105615, A107716, A107718-A107727.

Programs

  • PARI
    {T(n,k)=if(n
  • PARI
    {T(n,k)=if(n=j,if(m==j,1,if(m==j+1,-3*j,-T(m-j-1,0)))))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))

Formula

T(n, k) = 3*(k+1)*T(n, k+1) + Sum_{j=1..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>=0, with T(n, n) = 1 for n>=0. T(n, 0) = A107716(n+1) for n>=0.

A105629 Triangular matrix, read by rows, equal to the matrix logarithm of triangle A105623.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 17, 7, 3, 0, 135, 43, 13, 4, 0, 1353, 361, 93, 21, 5, 0, 16251, 3779, 883, 175, 31, 6, 0, 226857, 47077, 10277, 1893, 297, 43, 7, 0, 3605775, 678443, 140743, 24735, 3631, 467, 57, 8, 0, 64288209, 11095201, 2211413, 376209, 52961, 6385, 693
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

Also equals (1/2) the matrix logarithm of triangle A105615, since A105623 equals the matrix square-root of triangle A105615.

Examples

			Triangle begins:
0;
1,0;
3,2,0;
17,7,3,0;
135,43,13,4,0;
1353,361,93,21,5,0;
16251,3779,883,175,31,6,0;
226857,47077,10277,1893,297,43,7,0;
3605775,678443,140743,24735,3631,467,57,8,0;
64288209,11095201,2211413,376209,52961,6385,693,73,9,0; ...

Crossrefs

Cf. A105615, A105623, A105630 (column 0), A105631 (row sums).

Programs

  • PARI
    T(n,k)=local(L,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-1); L=sum(i=1,#M,(-1)^(i-1)*(M-M^0)^i/i); return(if(n

A105627 Column 1 of triangle A105626.

Original entry on oeis.org

1, 6, 48, 480, 5736, 79584, 1256808, 22262640, 437315016, 9438589824, 222109617288, 5661445534800, 155427839133096, 4573268363775264, 143592923776842408, 4792636497324986160, 169456698405536983176
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

A105626 equals the cube of the matrix square-root of triangle A105615.

Crossrefs

Cf. A105615, A105626.

Programs

  • PARI
    {a(n)=local(R,M=matrix(n+2,n+2,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-3); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n<0,0,R[n+2,2]))}
Showing 1-10 of 17 results. Next

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

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