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.

A111819 Column 0 of the matrix logarithm (A111818) of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, -2, 2, 840, -76056, -158761104, 390564896784, 14713376473366656, -783793232940393380736, -571732910947761663424746240, 603368029500890443054004423520000, 8390120127886533420753746115877557580800
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Let q=4; the g.f. of column k of A078536^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

Examples

			A(x) = x - 2/2!*x^2 + 2/3!*x^3 + 840/4!*x^4 - 76056/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3! +
A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
Let G(x) be the g.f. of A111817 (column 1 of A078536), then
G(x) = 1 + 4*A(x) + 4^2*A(x)*A(4*x)/2! +
4^3*A(x)*A(4*x)*A(4^2*x)/3! +
4^4*A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...

Crossrefs

Cf. A078536 (triangle), A111817, A111818 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

Programs

  • PARI
    {a(n,q=4)=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(4^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.

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.

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.

A111848 Matrix log of triangle A111845, 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, 4, 4, 0, 56, 16, 16, 0, 1728, 224, 64, 64, 0, -45696, 6912, 896, 256, 256, 0, -159401472, -182784, 27648, 3584, 1024, 1024, 0, 387212983296, -637605888, -731136, 110592, 14336, 4096, 4096, 0, 14722642769657856, 1548851933184, -2550423552, -2924544, 442368, 57344, 16384, 16384, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2005

Keywords

Comments

Column k equals 4^k multiplied by column 0 (A111849) when ignoring zeros above the diagonal.

Examples

			Matrix log of A111845, with factorial denominators, begins:
0;
1/1!, 0;
4/2!, 4/1!, 0;
56/3!, 16/2!, 16/1!, 0;
1728/4!, 224/3!, 64/2!, 64/1!, 0;
-45696/5!, 6912/4!, 896/3!, 256/2!, 256/1!, 0; ...

Crossrefs

Cf. A111845 (triangle), A111849 (column 0), A111818 (variant).

Programs

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

Formula

T(n, k) = 4^k*T(n-k, 0) = 4^k*A111844(n-k) for n>=k>=0.
Showing 1-8 of 8 results.

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