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.

A111816 Column 0 of the matrix logarithm (A111815) of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, -1, -3, 150, 1236, -2555748, -64342116, 5885700899760, 442646611978752, -1737387344860364226240, -367706581563500487774720, 60788555325888838346137808787840, 34626906551623392401873575206240000, -237458311254822429335982538087618909465992960
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Let q=3; the g.f. of column k of A078122^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

Examples

			E.g.f.: A(x) = x - 1/2!*x^2 - 3/3!*x^3 + 150/4!*x^4 + 1236/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3! +
A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
Let G(x) be the g.f. of A078124 (column 1 of A078122), then
G(x) = 1 + 3*A(x) + 3^2*A(x)*A(3*x)/2! +
3^3*A(x)*A(3*x)*A(3^2*x)/3! +
3^4*A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...

Crossrefs

Cf. A078122 (triangle), A078124, A111815 (matrix log); A110505 (q=-1), A111814 (q=2), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

Programs

  • PARI
    {a(n,q=3)=local(A=x/(1-x+x*O(x^n)));for(i=1,n, A=x/(1-x)/(1+sum(j=1,n,prod(k=1,j,subst(A,x,q^k*x))/(j+1)!))); return(n!*polcoeff(A,n))}

Formula

E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(3^j*x)/(j+1).

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.

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.

A111843 Matrix log of triangle A111840, 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, 3, 3, 0, 27, 9, 9, 0, 486, 81, 27, 27, 0, 7776, 1458, 243, 81, 81, 0, -2423196, 23328, 4374, 729, 243, 243, 0, -97338996, -7269588, 69984, 13122, 2187, 729, 729, 0, 5883879500784, -292016988, -21808764, 209952, 39366, 6561, 2187, 2187, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2005

Keywords

Comments

Column k equals 3^k multiplied by column 0 (A111844) when ignoring zeros above the diagonal.

Examples

			Matrix log of A111840, with factorial denominators, begins:
0;
1/1!, 0;
3/2!, 3/1!, 0;
27/3!, 9/2!, 9/1!, 0;
486/4!, 81/3!, 27/2!, 27/1!, 0;
7776/5!, 1458/4!, 243/3!, 81/2!, 81/1!, 0;
-2423196/6!, 23328/5!, 4374/4!, 729/3!, 243/2!, 243/1!, 0;

Crossrefs

Cf. A111840 (triangle), A111844 (column 0), A111815 (variant), A111941 (q=-1), A111810 (q=2), A111848 (q=4).

Programs

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

Formula

T(n, k) = 3^k*T(n-k, 0) = 3^k*A111844(n-k) for n>=k>=0.
Showing 1-8 of 8 results.

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

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