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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.

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A156933 FP4 polynomials related to the o.g.f.s of the columns of the A156925 matrix.

Original entry on oeis.org

1, 1, 1, -11, 156, -627, 736, 591, -1116, -369, -6, 106, -2772, 76070, -1087552, 8632650, -40358780, 106452214, -99774996, -284430514, 1125952500, -1581820542, 737716032, 414532350, -357790500, -81870750, -1275750
Offset: 0

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Author

Johannes W. Meijer, Feb 20 2009

Keywords

Comments

For the matrix of the coefficients of the FP2 see A156925. The coefficients in the columns of this matrix are the powers of z^m, m=0, 1, 2, ... . The columns are numbered: 1, 2, 3, ... .
The GF4(z;m) generate the sequences of the z^m coefficients. The general structure of the GF4(z;m) is given below.
The FP4(z,m) in the numerator of the GF4(z;m) is a polynomial of a certain degree, let's say k4. The (k4+1) coefficients of this polynomial can be determined one by one by comparing the series expansion of the FP4(z,m) with the values of the powers of z^m in column (m+1). These values can be generated with the GF2 formulas, see A156925.
An appropriate name for the polynomials FP4(z;m) in the numerators of the GF4(z;m) seems to be the flower polynomials of the fourth kind because the zero patterns of these polynomials look like flowers. The zero patterns of the FP4 and the FP3, see A156927, resemble each other closely and look like the zero patterns of the FP1 and FP2.
The sequence of (k4+1) number of terms of the FP4(z;m) polynomials for m from 0 to 11 is 1, 2, 7, 17, 28, 44, 63, 83, 108, 136, 167, 199.

Examples

			The first few rows of the "triangle" of the FP4(z;m) coefficients are:
[1]
[1, 1]
[ -11, 156, -627, 736, 591, -1116, -369]
The first few FP4 polynomials are:
FP4(z; m=0) = 1
FP4(z; m=1) = (1+z)
FP4(z; m=2) = ( -11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6 )
Some GF4(z;m) are:
GF4(z;m=1) = z*(1+z)/((1-3*z)*(1-z)^4)
GF4(z;m=2) = z^2*(-11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6)/((1-z)^7*(1-3*z)^4*(1-5*z))

Crossrefs

Cf. A156920, A156921, A156925, A156927.
For the first few GF4's see A156934, A156935, A156936, A156937.
Row sums A156938.
For the polynomials in the denominators of the GF4(z;m) see A157705. - Johannes W. Meijer, Mar 07 2009

Formula

G.f.: GF4(z;m):= z^q*FP4(z;m) / Product_{k=0..m} (1-(2*m+1-(2*k))*z)^(3*k+1).

A156934 G.f. of the z^1 coefficients of the FP2 in the second column of the A156925 matrix.

Original entry on oeis.org

1, 8, 38, 144, 487, 1552, 4796, 14592, 44061, 132568, 398210, 1195280, 3586659, 10760992, 32284216, 96854144, 290564217, 871694760, 2615086750, 7845263120, 23535792671, 70607381808
Offset: 1

Views

Author

Johannes W. Meijer, Feb 20 2009

Keywords

Crossrefs

Cf. A156933
Equals second column of A156925
Other columns A156935, A156936, A156937

Formula

a(n)=6*a(n-1)-12*a(n-2)+10*a(n-3)-3*a(n-4)+2
a(n)=7*a(n-1)-18*a(n-2)+22*a(n-3)-13*a(n-4)+3*a(n-5)
a(n):= -1/6*n^3-n^2-7/3*n-9/4+9/4*3^n
G.f.: GF4(z;m=1) = z*(1+z)/((1-3*z)*(1-z)^4)

A156935 G.f. of the z^2 coefficients of the FP2 in the third column of the A156925 matrix.

Original entry on oeis.org

-11, -108, -425, 720, 25678, 255048, 1901298, 12339600, 73794315, 418519612, 2289264465, 12207642976, 63925371180, 330387702160, 1691454191604, 8600517968736, 43516016460465, 219401923888740
Offset: 2

Views

Author

Johannes W. Meijer, Feb 20 2009

Keywords

Crossrefs

Cf. A156933
Equals third column of A156925
Other columns A156934, A156936, A156937

Formula

a(n)=24*a(n-1)-254*a(n-2)+1568*a(n-3)-6291*a(n-4)+17296*a(n-5)- 33460*a(n-6)+45984*a(n-7)-44663*a(n-8)+29976*a(n-9)-13230*a(n-10)+3456*a(n-11)-405*a(n-12)
a(n)= 75/32+101/24*n-225/16*3^n+375/32*5^n+289/72*n^2-1/8*3^(1+n)*n^3-9/4*3^n*(n)^2- 15/2*3^n*n+19/8*n^3+ 61/72*n^4+1/6*n^5+1/72*n^6
G.f.: GF4(z;m=2) = z^2*(-11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6)/((1-z)^7*(1-3*z)^4*(1-5*z))

A156936 G.f. of the z^3 coefficients of the FP2 in the fourth column of the A156925 matrix.

Original entry on oeis.org

-6, -242, -7382, -130472, -1594852, -15166900, -119173924, -788897224, -4270968154, -15821839894, 13226522262, 1056215331024, 14319250065624, 147391347765784, 1340374086462424
Offset: 2

Views

Author

Johannes W. Meijer, Feb 20 2009

Keywords

Crossrefs

Cf. A156933.
Equals fourth column of A156925.
Other columns A156934, A156935, A156937.

Formula

a(n) = 58*a(n-1) - 1571*a(n-2) + 26428*a(n-3)- 309755*a(n-4) + 2689810*a(n-5) - 17964865*a(n-6) + 94564560*a(n-7) - 398823930*a(n-8) + 1362709780*a(n-9) - 3799420462*a(n-10) + 8679603176*a(n-11) - 16269149542*a(n-12) + 24993226196*a(n-13) - 31349144530*a(n-14) + 31885547728*a(n-15) - 26017270869*a(n-16) + 16759251378*a(n-17) - 8320633119*a(n-18) + 3068440380*a(n-19) - 790800975*a(n-20) + 127028250*a(n-21) - 9568125*a(n-22).
G.f.: GF4(z;m=3) = z^2*(-6 + 106*z - 2772*z^2 + 76070*z^3 - 1087552*z^4 + 8632650*z^5 - 40358780*z^6 + 106452214*z^7 - 99774996*z^8 - 284430514*z^9 + 1125952500*z^10 - 1581820542*z^11 + 737716032*z^12 + 414532350*z^13 - 357790500*z^14 - 81870750*z^15 - 1275750*z^16)/((1-z)^10*(1-3*z)^7*(1-5*z)^4*(1-7*z)).
Showing 1-4 of 4 results.

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