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 1-9 of 9 results.

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

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.

A105630 Column 0 of triangle A105629, which is the matrix logarithm of triangle A105623.

Original entry on oeis.org

0, 1, 3, 17, 135, 1353, 16251, 226857, 3605775, 64288209, 1270969971, 27603549057, 653517822615, 16755529944729, 462601460800491, 13685474246611737, 431948067953729055, 14489465807596684449, 514794897939436455651
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

A105623 equals the matrix square-root of triangle A105615.

Crossrefs

Cf. A105629, A105623, A105615.

Programs

  • PARI
    {a(n)=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<0,0,L[n+1,1]/2))}

A105631 Row sums of triangle A105629, which is the matrix logarithm of triangle A105623.

Original entry on oeis.org

0, 1, 5, 27, 195, 1833, 21125, 286451, 4453859, 78031153, 1520668645, 32631020011, 764640901539, 19431070911513, 532315915981605, 15640217893829891, 490640673319698179, 16368499360721508833, 578693283962999831365
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

A105623 equals the matrix square-root of triangle A105615.

Crossrefs

Cf. A105629, A105623, A105615.

Programs

  • PARI
    {a(n)=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<0,0,sum(k=0,n,L[n+1,k+1])/2))}

A105624 Column 2 of triangle A105623.

Original entry on oeis.org

1, 3, 19, 167, 1831, 23843, 358339, 6097607, 115840951, 2430329603, 55812650419, 1392737182247, 37528377195271, 1086086115808163, 33600297897404899, 1106632150406054087, 38659524289511283991, 1427864216163041265923
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

A105623 equals the matrix square-root of triangle A105615.

Crossrefs

Cf. A105615, A105623.

Programs

  • PARI
    {a(n)=local(R,M=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); 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+3,3]))}

A105625 Row sums of triangle A105623.

Original entry on oeis.org

1, 2, 7, 40, 324, 3336, 41192, 590480, 9623944, 175703056, 3552295752, 78802665120, 1903505233064, 49743146641616, 1398474578414632, 42092742475096960, 1350629892258170184, 46025643554111478576
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

A105623 equals the matrix square-root of triangle A105615.

Crossrefs

Cf. A105615, A105623.

Programs

  • PARI
    {a(n)=local(R,M=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); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n<0,0,sum(k=0,n,R[n+1,k+1])))}

A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2*p-1)/2)) = (2*p-1)*(column p of T), or [T^((2*p-1)/2)](m,0) = (2*p-1)*T(p+m,p+1) for all m>=1 and p>=0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 74, 26, 6, 1, 706, 226, 50, 8, 1, 8162, 2426, 522, 82, 10, 1, 110410, 30826, 6498, 1010, 122, 12, 1, 1708394, 451586, 93666, 14458, 1738, 170, 14, 1, 29752066, 7489426, 1532970, 235466, 28226, 2754, 226, 16, 1, 576037442
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

Column 0 is A000698 (related to double factorials), offset 1. Column 1 is A105616 (column 0 of T^(1/2), offset 1). The matrix logarithm divided by 2 yields the integer triangle A105629.
Compare with triangular matrix A107717, which satisfies: SHIFT_LEFT(column 0 of A107717^((3*k-1)/3)) = (3*k-1)*(column k of A107717).

Examples

			SHIFT_LEFT(column 0 of T^(-1/2)) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^(1/2)) = 1*(column 1 of T);
SHIFT_LEFT(column 0 of T^(3/2)) = 3*(column 2 of T);
SHIFT_LEFT(column 0 of T^(5/2)) = 5*(column 3 of T).
Triangle begins:
1;
2,1;
10,4,1;
74,26,6,1;
706,226,50,8,1;
8162,2426,522,82,10,1;
110410,30826,6498,1010,122,12,1;
1708394,451586,93666,14458,1738,170,14,1;
29752066,7489426,1532970,235466,28226,2754,226,16,1; ...
Matrix square-root T^(1/2) is A105623 which begins:
1;
1,1;
4,2,1;
26,10,3,1;
226,74,19,4,1;
2426,706,167,31,5,1; ...
compare column 0 of T^(1/2) to column 1 of T;
also, column 1 of T^(1/2) equals column 0 of T.
Matrix inverse square-root T^(-1/2) is A105620 which begins:
1;
-1,1;
-2,-2,1;
-10,-4,-3,1;
-74,-20,-7,-4,1;
-706,-148,-39,-11,-5,1; ...
compare column 0 of T^(-1/2) to column 0 of T.
Matrix inverse T^-1 is A105619 which 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; ...

Crossrefs

Cf. A000698 (column 0), A105616 (column 1), A105617 (column 2), A105618 (row sums), A105619 (T^-1), A105620 (T^(-1/2)), A105623 (T^(1/2)), A105627 (T^(3/2)), A105629 (matrix log).
Cf. A107717.

Programs

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

Formula

T(n, k) = 2*(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) = A000698(n+1) for n>=0.

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])}
Showing 1-9 of 9 results.

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

Content is available under The OEIS End-User License Agreement.