A156920 -id:A156920 - cp's OEIS frontend

A156927 FP3 polynomials related to the generating functions of the columns of the A156921 matrix.

1, 1, 1, -6, 29, 31, -283, 245, 298, -286, -108, 119, -3106, 29469, -104585, -220481, 3601363, -15487305, 34949165, -39821950, 4356011, 46881744, -51274736, 9005908, 14663472, -5205168, -1456704, -20736

Offset: 0

Johannes W. Meijer, Feb 20 2009

For the matrix of the FP1 polynomials see A156921. 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 GF3(z;m) generate the sequences of the z^m coefficients. The general structure of the GF3(z;m) is given below.
The FP3(z,m) in the numerator of the GF3(z;m) is a polynomial of a certain degree, let's say k3. The (k3+1) coefficients of this polynomial can be determined one by one by comparing the series expansion of the FP3(z,m) with the values of the powers of z^m in column (m+1). These values can be generated with the GF1 formulas, see A156921.
An appropriate name for the polynomials FP3(z;m) in the numerators of the GF(3;m) seems to be the flower polynomials of the third kind, the FP3, because the zero patterns of these polynomials look like flowers. The zero patterns of the FP3 and the FP4, see A156933, resemble each other closely and look like the zero patterns of the FP1 and FP2.
The sequence of the (k3+1) number of terms of the FP3(z;m) polynomials for m from 0 to 11 is 1, 2, 8, 17, 29, 45, 63, 84, 109, 137, 167, 200.

			The first few rows of the "triangle" of the FP3(z,m) coefficients are:
  [1]
  [1, 1]
  [-6, 29, 31, -283, 245, 298, -286, -108]
The first few FP3 polynomials are:
  FP3(z; m=0) = 1
  FP3(z; m=1) = (1+z)
  FP3(z; m=2) = (-6+29*z+31*z^2-283*z^3+245*z^4+298*z^5-286*z^6-108*z^7)
Some GF3(z;m) are:
  GF3(z;m=1) = z^2*(1+z)/((1-z)^4*(1-2*z))
  GF3(z;m=2) = z^2*(-6+29*z+31*z^2-283*z^3+245*z^4+298*z^5-286*z^6-108*z^7)/((1-z)^7*(1-2*z)^4*(1-3*z))

Cf. A156920, A156921, A156925, A156933.
For the first few GF3's see A156928, A156929, A156930, A156931.
Row sums A156932.
For the polynomials in the denominators of the GF3(z;m) see A157704.

G.f.: GF3(z;m):= z^p*FP3(z;m)/Product_{k=0..m} (1-(k+1)*z)^(1+3*k).

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

Johannes W. Meijer, Feb 20 2009

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.

			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))

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

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

A142965 One fourth of third column (m=2) of triangle A142963.

Original entry on oeis.org

1, 18, 129, 646, 2685, 10002, 34777, 115566, 372453, 1175290, 3654369, 11245110, 34349005, 104373282, 315969705, 954002878, 2874983541, 8652474378, 26015617585, 78169534470, 234766551261, 704840716978, 2115654610809, 6349329417486, 19052920751365, 57169029907482

Offset: 0

Wolfdieter Lang, Sep 15 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -40, 82, -91, 52, -12).

Column m=1: 2*A142964; m=3: 8*A142966.
From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156925.
Equals A156920(n+2,2).
Equals A156919(n+2,2)/2^n.
(End)

[35/2+2*n^2+12*n-84*2^n-24*2^n*n+135/2*3^n: n in [0..25]]; // Vincenzo Librandi, Jun 18 2017

LinearRecurrence[{10,-40,82,-91,52,-12}, {1,18,129,646,2685,10002},30] (* or *) CoefficientList[Series[(1+8x-11x^2-6x^3)/((x-1)^3 (2x-1)^2 (3x-1)),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2011 *)

a(n) = A142963(n+3,2)/4.
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n) = 10a(n-1) - 40a(n-2) + 82a(n-3) - 91a(n-4) + 52a(n-5) - 12a(n-6).
a(n) = 35/2 + 2*n^2 + 12*n - 84*2^n - 24*2^n*n + 135/2*3^n
G.f.: (1 + 8*z - 11*z^2 - 6*z^3)/((1-z)^3*(1-2*z)^2*(1-3*z)).
(End)

A142966 Fourth column (m=3) of triangle A142963 divided by 8.

Original entry on oeis.org

1, 58, 877, 8030, 56285, 335162, 1792749, 8904486, 41949645, 190129090, 837258109, 3607669966, 15289404989, 63975698570, 265065915725, 1089837752118, 4454225465325, 18119738464530, 73441531708765, 296814738679390, 1196884383319261, 4817845684107098

Offset: 0

Wolfdieter Lang, Sep 15 2008

Cf. A142963.
Column m=2: 4*A142965.
From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156925.
Equals A156920(n+3,3).
Equals A156919(n+3,3)/2^n.
(End)

a(n) = A142963(n+4,3)/8.
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n) = 20a(n-1) - 175*a(n-2) + 882*a(n-3) - 2835*a(n-4) + 6072*a(n-5) - 8777*a(n-6) + 8458*a(n-7) - 5204*a(n-8) + 1848*a(n-9) - 288*a(n-10).
a(n) = -(105/2) - (143/3)*n - 14*n^2 - (4/3)*n^3 + 756*2^n + 48*2^n*n^2 + 384*2^n*n - (3645/2)*3^n - 405*3^n*n + 1120*4^n.
G.f.: (1 + 38*z - 108*z^2 - 242*z^3 + 839*z^4 - 444*z^5 - 180*z^6)/((1-z)^4*(1-2*z)^3*(1-3*z)^2*(1-4*z)).
(End)

A142968 Fifth column (m=4) of triangle A142963 divided by 16=2^4.

Original entry on oeis.org

1, 179, 5280, 82610, 919615, 8284857, 64730022, 457217400, 2999230965, 18608607535, 110625457964, 636103699038, 3562753619915, 19541111960965, 105392471360850, 560747327119908, 2950726075955265, 15387821226034875, 79656442803398680, 409857988825489610

Offset: 0

Wolfdieter Lang, Sep 15 2008

Column m=3: 8*A142966.
From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156925.
Equals A156920(n+4,4).
Equals A156919(n+4,4)/2^n.
(End)

a(n) = A142963(n+5,3)/2^4.
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n) = 35a(n-1) - 560a(n-2) + 5432a(n-3) - 35714a(n-4) + 168542a(n-5) - 589632a(n-6) + 1556776a(n-7) - 3126949a(n-8) + 4777591a(n-9) - 5506936a(n-10) + 4703032a(n-11) - 2881136a(n-12) + 1195632a(n-13) - 300672a(n-14) + 34560a(n-15).
a(n) = (1155/8) + (472/3)*n - 5544*2^n + (120285/4)*3^n - 49280*4^n + (196875/8)*5^n - 64*2^n*n^3 - 864*2^n*n^2 - 3824*2^n*n + (187/3)*n^2 + 1215*3^n*n^2 + 12150*3^n*n - 8960*4^n*n + (32/3)*n^3 + (2/3)*n^4.
G.f.: (1 + 144*z - 425*z^2 - 7382*z^3 + 48451*z^4 - 96764*z^5 - 2559*z^6 + 257002*z^7 - 312444*z^8 + 88344*z^9 + 30240*z^10)/((1-z)^5*(1-2*z)^4*(1-3*z)^3*(1-4*z)^2*(1-5*z)).
(End)

A142964 a(n) = 62^n - 2n - 5.

Original entry on oeis.org

1, 5, 15, 37, 83, 177, 367, 749, 1515, 3049, 6119, 12261, 24547, 49121, 98271, 196573, 393179, 786393, 1572823, 3145685, 6291411, 12582865, 25165775, 50331597, 100663243, 201326537, 402653127, 805306309, 1610612675, 3221225409, 6442450879, 12884901821

Offset: 0

Wolfdieter Lang, Sep 15 2008

Previous name was: One half of second column (m=1) of triangle A142963.

Essentially a duplicate of A050488. - Johannes W. Meijer, Feb 20 2009

			a(3) = 6*2^3 - 2*3 - 5 = 37.

Eric Billault, Walter Damin, Robert Ferréol, and Rodolphe Garin, MPSI Classes Prépas - Khôlles de Maths, Exercices corrigés, Ellipses, 2012, exercice 2.22 (1) pp 26, 43-44.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A142965 (m=2 column/4).
Equals A050488(n+1).
Equals A156920(n+1,1).
Equals A156919(n+1,1)/2^n.
Cf. A156925, A339771.
Partial sums of A033484.

seq(6*2^n-2*n-5,n=0..40); # Bernard Schott, Dec 16 2020

a[n_]:=6*2^n-2n-5;Array[a,32,0] (* or *) CoefficientList[Series[(1+x)/((1-x)^2*(1-2*x)),{x,0,31}],x] (* or *) LinearRecurrence[{4,-5,2},{1,5,15},32] (* James C. McMahon, Aug 12 2025 *)

Vec((1+z)/((1-z)^2*(1-2*z)) + O(z^50)) \\ Michel Marcus, Jun 18 2017

a(n) = A142693(n+2,1)/2.
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n) = 4a(n-1) - 5a(n-2) + 2a(n-3) for n > 2 with a(0) = 1, a(1) = 5, a(2) = 15.
G.f.: (1+z)/((1-z)^2*(1-2*z)). (End)
a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^min(i,j) (Billault et al) (compare with A339771 that has max instead of min). - Bernard Schott, Dec 16 2020
a(n) = 2*A066524(n+1) - A339771(n). - Kevin Ryde, Dec 17 2020
E.g.f.: 6*exp(2*x) - exp(x)*(5 + 2*x). - Stefano Spezia, Dec 17 2020

New name using a formula of Bernard Schott by Peter Luschny, Dec 17 2020

A211399 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 15, 18, 1, 0, 1, 37, 129, 58, 1, 0, 1, 83, 646, 877, 179, 1, 0, 1, 177, 2685, 8030, 5280, 543, 1, 0, 1, 367, 10002, 56285, 82610, 29658, 1636, 1, 0, 1, 749, 34777, 335162

Offset: 0

Philippe Deléham, Feb 08 2013

Contains A156920 as submatrix.

Row-reversal of A102365. - Philippe Deléham, Feb 12 2013

			Triangle begins :
1
0, 1
0, 1, 1
0, 1, 5, 1
0, 1, 15, 18, 1
0, 1, 37, 129, 58, 1
0, 1, 83, 646, 877, 179, 1

Left hand column sequences: A000007, A000012, A050488, A142965, A142966, A142968.
Right hand column sequences: A000340, A156922, A156923, A156924.
Row sums A014307(n).

T(n,k) = k*T(n-1,k) + (2n-2k+1)*T(n-1,k-1) , T(n,n) = 1, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A185411(n,k)/(2^(n-k)).
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A014307(n), A001147(n) for x = 0, 1, 2 respectively .
G.f.: 1/(1-xy/(1-x/(1-3xy/(1-2x/(1-5xy/(1-3x/(1-7xy/(1- ...(continued fraction).

cp's OEIS Frontend

A156927 FP3 polynomials related to the generating functions of the columns of the A156921 matrix.

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

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A142965 One fourth of third column (m=2) of triangle A142963.

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A142966 Fourth column (m=3) of triangle A142963 divided by 8.

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A142968 Fifth column (m=4) of triangle A142963 divided by 16=2^4.

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A142964 a(n) = 6*2^n - 2*n - 5.

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A211399 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) where DELTA is the operator defined in A084938.

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A142964 a(n) = 62^n - 2n - 5.