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

Original entry on oeis.org

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

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Author

Johannes W. Meijer, Feb 20 2009

Keywords

Comments

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.

Examples

			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))

Crossrefs

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.

Formula

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

A156928 G.f. of the z^1 coefficients of the FP1 in the second column of the A156921 matrix.

Original entry on oeis.org

1, 7, 28, 86, 227, 545, 1230, 2664, 5613, 11611, 23728, 48106, 97031, 195077, 391394, 784284, 1570353, 3142815, 6288100, 12579070, 25161451, 50326697, 100657718, 201320336, 402646197, 805298595
Offset: 2

Views

Author

Johannes W. Meijer, Feb 20 2009

Keywords

Comments

Antidiagonal sums of the convolution array A213582. - Clark Kimberling, Jun 19 2012

Links

Crossrefs

Cf. A156927.
Equals second column of A156921.
Other columns A156929, A156930, A156931.

Programs

  • GAP
    List([2..40], n-> (9*2^(n+2) -(2*n^3+9*n^2+25*n+36))/6); # G. C. Greubel, Jul 08 2019
  • Magma
    [(9*2^(n+2) -(2*n^3+9*n^2+25*n+36))/6: n in [2..40]]; // G. C. Greubel, Jul 08 2019
  • Mathematica
    Table[(9*2^(n+2) -(2*n^3+9*n^2+25*n+36))/6, {n, 2, 40}] (* Michael De Vlieger, Sep 23 2017 *)
  • PARI
    vector(40, n, n++; (9*2^(n+2) -(2*n^3+9*n^2+25*n+36))/6) \\ G. C. Greubel, Jul 08 2019
  • Sage
    [(9*2^(n+2) -(2*n^3+9*n^2+25*n+36))/6 for n in (2..40)] # G. C. Greubel, Jul 08 2019

Formula

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) + 2.
a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
a(n) = (9*2^(n+2) - (2*n^3 + 9*n^2 + 25*n + 36))/6.
G.f.: GF3(z;m=1) = z^2*(1+z)/((1-z)^4*(1-2*z)).
a(n) = Sum_{k=1..n+1} Sum_{i=1..n+1} (k-1)^2 * C(n-k+1,i). - Wesley Ivan Hurt, Sep 22 2017
E.g.f.: (36*exp(2*x) - (36 + 36*x + 15*x^2 + 2*x^3)*exp(x))/6. - G. C. Greubel, Jul 08 2019

A156930 G.f. of the z^3 coefficients of the FP1 in the fourth column of the A156921 matrix.

Original entry on oeis.org

119, 1654, 5784, -57421, -974120, -8191112, -51264392, -266367722, -1204269647, -4832097594, -17187443308, -52433219783, -120342975558, -58288009528, 1603731045044, 13940518848356
Offset: 3

Views

Author

Johannes W. Meijer, Feb 20 2009

Keywords

Crossrefs

Cf. A156927
Equals fourth column of A156921
Other columns A156928, A156929, A156931

Formula

a(n)=40*a(n-1)-755*a(n-2)+8946*a(n-3)-74677*a(n-4)+467156*a(n-5)-2274363*a(n-6)+8833486*a(n-7)-27833039*a(n-8)+71958408*a(n-9)-153781873*a(n-10)+272810702*a(n-11)-402324879*a(n-12)+492639700*a(n-13)-498877265*a(n-14)+414825042*a(n-15)-280100140*a(n-16)+151065320*a(n-17)-63500432*a(n-18)+20037984*a(n-19)-4463424*a(n-20)+625536*a(n-21)-41472*a(n-22)
G.f.: GF3(z;m=3) = z^3*( 119-3106*z+29469*z^2-104585*z^3-220481*z^4+3601363*z^5-15487305*z^6+34949165*z^7-39821950*z^8+4356011*z^9+46881744*z^10-51274736*z^11+ 9005908*z^12+14663472*z^13-5205168*z^14-1456704*z^15-20736*z^16)/((1-z)^10*(1-2*z)^7*(1-3*z)^4*(1-4*z))

A156931 G.f. of the z^4 coefficients of the FP1 in the fifth column of the A156921 matrix.

Original entry on oeis.org

126, 8689, 300930, 5663483, 69028169, 613038531, 4234224501, 23275739871, 98332765273, 250304681662, -554375755759, -13379311589392, -119762221369238, -826135093245122, -4949174987335110
Offset: 3

Views

Author

Johannes W. Meijer, Feb 20 2009

Keywords

Crossrefs

Cf. A156927
Equals fifth column of A156921
Other columns A156928, A156929, A156930

Formula

G.f.: GF3(z;m=4) = z^3*(126-761*z-9285*z^2-1277673*z^3+46183284*z^4-696765279*z^5+5823518147*z^6-25089694147*z^7-12711864696*z^8+988743755109*z^9-7652560832135*z^10+35510543219541*z^11- 113638922483756*z^12+ 252348200449359*z^13-345027066386175*z^14+ 81356029034411*z^15+ 881468053442834*z^16- 2383892701491756*z^17+3430328807256360*z^18-2918614790283444*z^19+ 1018849254500448*z^20+712292304270640*z^21-1071408562243680*z^22+ 470620280404224*z^23+19240926537600*z^24-78111305206272*z^25+ 11628096196608*z^26+3657667166208*z^27+99851304960*z^28 )/((1-z)^13*(1-2*z)^10*(1-3*z)^7*(1-4*z)^4*(1-5*z))
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