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 10 results.

A110505 Numerators of unsigned columns of triangle A110504: a(n) = n!*A110504(n,0) = (-1)^k*n!*A110504(n+k,k) for all k >= 0.

Original entry on oeis.org

0, 1, 3, 7, 30, 144, 876, 6084, 48816, 438624, 4389120, 48263040, 579242880, 7529552640, 105417365760, 1581231456000, 25299906508800, 430096581734400, 7741753102540800, 147093162635059200, 2941864569520128000
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2005

Keywords

Comments

Triangle A110504 equals the matrix logarithm of triangle A110503.
Triangle A110503 shifts one column left under matrix inverse.
Lim_{n->infinity} a(n)/n! = Pi*2*sqrt(3)/9 = 1.209199576...

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 7*x^3/3! + 30*x^4/4! + 144*x^5/5! + 876*x^6/6! + ...
where A(x) satisfies: A(x)*A(-x) = -arccos(1-1/2*x^2)^2, and
arccos(1-1/2*x^2)^2 = Sum_{n>=0} x^(2*n+2)/( C(2*n+1, n)*(n+1)^2 ) = x^2 + 1/12*x^4 + 1/90*x^6 + 1/560*x^8 + 1/3150*x^10 + ...

Links

Crossrefs

Cf. A110503 (triangular matrix), A110504 (matrix logarithm), A002544.

Programs

  • PARI
    /* From relation to unsigned columns of triangle A110504: */
{a(n)=local(M=matrix(n+1,n+1,r,c,if(r>=c, if(r==c || c%2==1,1,if(r%2==0 && r==c+2,-2,-1))))); n!*sum(i=1,#M,-(M^0-M)^i/i)[n+1,1]}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* As Partial Sums of Series: */
a(n)=if(n<1,0,n!*(1+sum(n=2,n,(-1)^n/(binomial(n-2,n\2-1)*n*(n-1)/((n+1)\2)))))
for(n=0,30,print1(a(n),", "))

Formula

E.g.f.: (2+x-x^2)/(2*(1-x)) * arccos(1-x^2/2) / sqrt(1-x^2/4).
E.g.f. A(x) satisfies:
(1) A(x)*A(-x) = -arccos(1-1/2*x^2)^2 = Sum_{n>=0} -x^(2*n+2)/( C(2*n+1, n)*(n+1)^2 ).
(2) 1/(1-x) = Sum_{n>=1} A(x)^floor((n+1)/2) * A(-x)^floor(n/2)/n!.
a(2*n+1) = (2*n+1)!*(1 + Sum_{k=1..n} (1/binomial(2*k+1, k))/(k+1)).
a(2*n+2) = (2*n+2)!*(1 + 1/2 - Sum_{k=1..n} 1/binomial(2*k+2, k)/k) = n!*(1 + 1/2 - 1/3 + 1/12 - 1/20 + 1/60 - 1/105 + 1/280 -+ ...).
Recurrence: 4*a(n) = 2*(2*n-1)*a(n-1) + (n-2)*(n+1)*a(n-2) - (n-3)*(n-2)*n*a(n-3). - Vaclav Kotesovec, May 09 2014

A110503 Triangle, read by rows, which shifts one column left under matrix inverse.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -2, 1, 1, 1, -1, 1, -1, 1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2005

Keywords

Comments

The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.

Examples

			Triangle begins:
  1;
  1,  1;
  1, -1,  1;
  1, -2,  1,  1;
  1, -1,  1, -1,  1;
  1, -1,  1, -2,  1,  1;
  1, -1,  1, -1,  1, -1,  1;
  1, -1,  1, -1,  1, -2,  1,  1;
  1, -1,  1, -1,  1, -1,  1, -1,  1;
  1, -1,  1, -1,  1, -1,  1, -2,  1,  1; ...
The matrix inverse drops the first column:
   1;
  -1,  1;
  -2,  1,  1;
  -1,  1, -1,  1;
  -1,  1, -2,  1,  1;
  -1,  1, -1,  1, -1,  1; ...
The matrix logarithm equals:
     0;
    1/1!,     0;
    3/2!,   -1/1!,   0;
    7/3!,   -3/2!,  1/1!,   0;
   30/4!,   -7/3!,  3/2!, -1/1!,  0;
  144/5!,  -30/4!,  7/3!, -3/2!, 1/1!,   0;
  876/6!, -144/5!, 30/4!, -7/3!, 3/2!, -1/1!, 0; ...
unsigned columns of which all equal A110505.

Crossrefs

Cf. A110504 (matrix log), A110505 (column 0 of log).
Cf. A111940 (variant).

Programs

  • PARI
    T(n,k)=matrix(n+1,n+1,r,c,if(r>=c, if(r==c || c%2==1,1,if(r%2==0 && r==c+2,-2,-1))))[n+1,k+1]

Formula

T(n, k) = +1 when k == 0 (mod 2), T(n, k)=-1 when k == 1 (mod 2), except for T(k+2, k) = -2 when k == 1 (mod 2) and T(n, n) = 1.
G.f. for column k of matrix power A110503^m (ignoring leading zeros): cos(m*arccos(1-x^2/2)) + (-1)^k*sin(m*arccos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4)*(1+x)/(1-x).

A111813 Matrix log of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, -2, 0, 4, 0, 0, -4, 0, 8, 0, 216, 0, -8, 0, 16, 0, 0, 432, 0, -16, 0, 32, 0, -568464, 0, 864, 0, -32, 0, 64, 0, 0, -1136928, 0, 1728, 0, -64, 0, 128, 0, 36058658688, 0, -2273856, 0, 3456, 0, -128, 0, 256, 0, 0, 72117317376, 0, -4547712, 0, 6912, 0, -256, 0, 512, 0
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 2^k multiplied by column 0 (A111814) when ignoring zeros above the diagonal.

Examples

			Matrix log of A078121, with factorial denominators, begins:
0;
1/1!, 0;
0/2!, 2/1!, 0;
-2/3!, 0/2!, 4/1!, 0;
0/4!, -4/3!, 0/2!, 8/1!, 0;
216/5!, 0/4!, -8/3!, 0/2!, 16/1!, 0;
0/6!, 432/5!, 0/4!, -16/3!, 0/2!, 32/1!, 0;
-568464/7!, 0/6!, 864/5!, 0/4!, -32/3!, 0/2!, 64/1!, 0; ...

Crossrefs

Cf. A078121, A111814 (column 0), A111810 (variant); log matrices: A110504 (q=-1), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

Programs

  • PARI
    T(n,k,q=2)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 2^k*T(n-k, 0) = A111814(n-k) for n>=k>=0.

A111815 Matrix log of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -1, 3, 0, -3, -3, 9, 0, 150, -9, -9, 27, 0, 1236, 450, -27, -27, 81, 0, -2555748, 3708, 1350, -81, -81, 243, 0, -64342116, -7667244, 11124, 4050, -243, -243, 729, 0, 5885700899760, -193026348, -23001732, 33372, 12150, -729, -729, 2187, 0
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 3^k multiplied by column 0 (A111816) when ignoring zeros above the diagonal.

Examples

			Matrix log of A078122, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, 3/1!, 0;
-3/3!, -3/2!, 9/1!, 0;
150/4!, -9/3!, -9/2!, 27/1!, 0;
1236/5!, 450/4!, -27/3!, -27/2!, 81/1!, 0;
-2555748/6!, 3708/5!, 1350/4!, -81/3!, -81/2!, 243/1!, 0; ...

Crossrefs

Cf. A078122, A111816 (column 0), A111840 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

Programs

  • PARI
    T(n,k,q=3)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 3^k*T(n-k, 0) = A111816(n-k) for n>=k>=0.

A111818 Matrix log of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -2, 4, 0, 2, -8, 16, 0, 840, 8, -32, 64, 0, -76056, 3360, 32, -128, 256, 0, -158761104, -304224, 13440, 128, -512, 1024, 0, 390564896784, -635044416, -1216896, 53760, 512, -2048, 4096, 0, 14713376473366656, 1562259587136, -2540177664, -4867584, 215040, 2048, -8192, 16384, 0
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 4^k multiplied by column 0 (A111819) when ignoring zeros above the diagonal.

Examples

			Matrix log of A078536, with factorial denominators, begins:
0;
1/1!, 0;
-2/2!, 4/1!, 0;
2/3!, -8/2!, 16/1!, 0;
840/4!, 8/3!, -32/2!, 64/1!, 0;
-76056/5!, 3360/4!, 32/3!, -128/2!, 256/1!, 0;
-158761104/6!, -304224/5!, 13440/4!, 128/3!, -512/2!, 1024/1!, 0;

Crossrefs

Cf. A078536, A111819 (column 0), A111845 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

Programs

  • PARI
    T(n,k,q=4)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 4^k*T(n-k, 0) = A111819(n-k) for n>=k>=0.

A111823 Matrix log of triangle A111820, which shifts columns left and up under matrix 5th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -3, 5, 0, 16, -15, 25, 0, 2814, 80, -75, 125, 0, -1092180, 14070, 400, -375, 625, 0, -3603928080, -5460900, 70350, 2000, -1875, 3125, 0, 58978973128440, -18019640400, -27304500, 351750, 10000, -9375, 15625, 0, 5974833380453777520
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 5^k multiplied by column 0 (A111824) when ignoring zeros above the diagonal.

Examples

			Matrix log of A111820, with factorial denominators, begins:
0;
1/1!, 0;
-3/2!, 5/1!, 0;
16/3!, -15/2!, 25/1!, 0;
2814/4!, 80/3!, -75/2!, 125/1!, 0;
-1092180/5!, 14070/4!, 400/3!, -375/2!, 625/1!, 0; ...

Crossrefs

Cf. A111820, A111824 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111828 (q=6), A111833 (q=7), A111838 (q=8).

Programs

  • PARI
    T(n,k,q=5)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 5^k*T(n-k, 0) = A111824(n-k) for n>=k>=0.

A111828 Matrix log of triangle A111825, which shifts columns left and up under matrix 6th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -4, 6, 0, 42, -24, 36, 0, 7296, 252, -144, 216, 0, -7931976, 43776, 1512, -864, 1296, 0, -45557382240, -47591856, 262656, 9072, -5184, 7776, 0, 3064554175021200, -273344293440, -285551136, 1575936, 54432, -31104, 46656, 0, 801993619807364206080
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 6^k multiplied by column 0 (A111829) when ignoring zeros above the diagonal.

Examples

			Matrix log of A111825, with factorial denominators, begins:
0;
1/1!, 0;
-4/2!, 6/1!, 0;
42/3!, -24/2!, 36/1!, 0;
7296/4!, 252/3!, -144/2!, 216/1!, 0;
-7931976/5!, 43776/4!, 1512/3!, -864/2!, 1296/1!, 0; ...

Crossrefs

Cf. A111825, A111829 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111833 (q=7), A111838 (q=8).

Programs

  • PARI
    T(n,k,q=6)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 6^k*T(n-k, 0) = A111829(n-k) for n>=k>=0.

A111833 Matrix log of triangle A111830, which shifts columns left and up under matrix 7th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -5, 7, 0, 83, -35, 49, 0, 16110, 581, -245, 343, 0, -40097784, 112770, 4067, -1715, 2401, 0, -388036363380, -280684488, 789390, 28469, -12005, 16807, 0, 82804198261002036, -2716254543660, -1964791416, 5525730, 199283, -84035, 117649, 0
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 7^k multiplied by column 0 (A111834) when ignoring zeros above the diagonal.

Examples

			Matrix log of A111830, with factorial denominators, begins:
0;
1/1!, 0;
-5/2!, 7/1!, 0;
83/3!, -35/2!, 49/1!, 0;
16110/4!, 581/3!, -245/2!, 343/1!, 0;
-40097784/5!, 112770/4!, 4067/3!, -1715/2!, 2401/1!, 0; ...

Crossrefs

Cf. A111830, A111834 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111838 (q=8).

Programs

  • PARI
    T(n,k,q=7)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 7^k*T(n-k, 0) = A111834(n-k) for n>=k>=0.

A111838 Matrix log of triangle A111835, which shifts columns left and up under matrix 8th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -6, 8, 0, 142, -48, 64, 0, 31800, 1136, -384, 512, 0, -159468264, 254400, 9088, -3072, 4096, 0, -2481298801008, -1275746112, 2035200, 72704, -24576, 32768, 0, 1414130111428687344, -19850390408064, -10205968896, 16281600, 581632, -196608, 262144, 0
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Column k equals 8^k multiplied by column 0 (A111839) when ignoring zeros above the diagonal.

Examples

			Matrix log of A111835, with factorial denominators, begins:
0;
1/1!, 0;
-6/2!, 8/1!, 0;
142/3!, -48/2!, 64/1!, 0;
31800/4!, 1136/3!, -384/2!, 512/1!, 0;
-159468264/5!, 254400/4!, 9088/3!, -3072/2!, 4096/1!, 0; ...

Crossrefs

Cf. A111835, A111839 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7).

Programs

  • PARI
    T(n,k,q=8)=local(A=Mat(1),B);if(n

Formula

T(n, k) = 8^k*T(n-k, 0) = A111839(n-k) for n>=k>=0.

A111941 Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -1, -1, 0, 1, 1, 1, 0, -2, -1, -1, -1, 0, 4, 2, 1, 1, 1, 0, -12, -4, -2, -1, -1, -1, 0, 36, 12, 4, 2, 1, 1, 1, 0, -144, -36, -12, -4, -2, -1, -1, -1, 0, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -86400, -14400, -2880, -576, -144, -36
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2005

Keywords

Examples

			Triangle begins:
0;
1, 0;
-1, -1, 0;
1, 1, 1, 0;
-2, -1, -1, -1, 0;
4, 2, 1, 1, 1, 0;
-12, -4, -2, -1, -1, -1, 0;
36, 12, 4, 2, 1, 1, 1, 0;
-144, -36, -12, -4, -2, -1, -1, -1, 0;
576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
518400, 86400, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-3628800, -518400, -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0; ...
where, apart from signs, the columns are all the same (A111942).
...
Triangle A111940 begins:
1;
1, 1;
-1, -1, 1;
0, 0, 1, 1;
0, 0, -1, -1, 1;
0, 0, 0, 0, 1, 1;
0, 0, 0, 0, -1, -1, 1;
0, 0, 0, 0, 0, 0, 1 ,1;
0, 0, 0, 0, 0, 0, -1, -1, 1; ...
where the matrix inverse shifts columns left and up one place.
...
The matrix log of A111940, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, -1/1!, 0;
1/3!, 1/2!, 1/1!, 0;
-2/4!, -1/3!, -1/2!, -1/1!, 0;
4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-2880/10!, -576/9!, -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
14400/11!, 2880/10!, 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; ...
Note that the square of the matrix log of A111940 begins:
0;
0, 0;
-1, 0, 0;
0, -1, 0, 0;
-1/12, 0, -1, 0, 0;
0, -1/12, 0, -1, 0, 0;
-1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/16632, 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; ...
where nonzero terms are negative unit fractions with denominators given by A002544:
[1, 12, 90, 560, 3150, 16632, 84084, 411840, ...,  C(2*n+1,n)*(n+1)^2, ...].

Crossrefs

Cf. A111940 (triangle), A111942 (column 0), A110504 (variant).

Programs

  • PARI
    {T(n,k,q=-1) = local(A=Mat(1),B); if(n

Formula

T(n, k) = (-1)^k*T(n-k, 0) = (-1)^k*A111942(n-k) for n>=k>=0.
Showing 1-10 of 10 results.

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