A008615 -id:A008615 - cp's OEIS frontend

A083024 Molien series for action of SL(3,C) on ternary forms of degree 4.

1, 1, 2, 4, 7, 11, 19, 29, 44, 67, 98, 139, 199, 275, 375, 509, 678, 890, 1165, 1501, 1916, 2431, 3053, 3801, 4711, 5788, 7063, 8580, 10353, 12420, 14841, 17633, 20850, 24565, 28807, 33641, 39161, 45404, 52455, 60427, 69372, 79392, 90627, 103143, 117065, 132561

Offset: 0

Benoit Cloitre, Jun 01 2003

These are the coefficients of the expansion in powers of z^4, the other coefficients being zero.

J-M. Kantor, Où en sont les mathématiques. La formule de Molien-Weyl, SMF, Vuibert, p. 79

T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
Index entries for Molien series.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,2,0,0,1,0,-1,0,1,0,-1,0,0,-2,0,2,0,1,0,0,-1,-1,1).

Cf. A008615.

seq(coeff(series( (1 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 3*x^11 + 4*x^12 + 4*x^13 + 3*x^14 + 4*x^15 + 3*x^16 + 2*x^17 + 3*x^18 + 2*x^19 + x^20 + x^21 + x^25)/(1 - x^1 - x^2 + x^5 + 2*x^7 - 2*x^9 - x^12 + x^14 - x^16 + x^18 + 2*x^21 - 2*x^23 - x^25 + x^28 + x^29 - x^30), x, n+1), x, n), n = 0..45); # Georg Fischer, Jan 24 2021

a(n)=polcoeff((1+z^9+z^12+z^15+2*z^18+3*z^21+2*z^24+3*z^27+4*z^30+3*z^33 +4*z^36+4*z^39+3*z^42+4*z^45+3*z^48+2*z^51+3*z^54+2*z^57+z^60+z^63+z^75) /(1-z^3)/(1-z^6)/(1-z^9)/(1-z^12)/(1-z^15)/(1-z^18)/(1- z^27)+O(z^(n+1)),n)

G.f.: (1 + z^9 + z^12 + z^15 + 2*z^18 + 3*z^21 + 2*z^24 + 3*z^27 + 4*z^30 + 3*z^33 + 4*z^36 + 4*z^39 + 3*z^42 + 4*z^45 + 3*z^48 + 2*z^51 + 3*z^54 + 2*z^57 + z^60 + z^63 + z^75)/(1-z^3)/(1-z^6)/(1-z^9)/(1-z^12)/(1-z^15)/(1-z^18)/(1-z^27).

A165686 Dimension of the space of Siegel cusp forms of genus 2 and weight 2k which are not Saito-Kurokawa lifts of forms of genus 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 8, 11, 12, 14, 16, 19, 20, 24, 26, 29, 32, 37, 38, 44, 47, 51, 56, 62, 64, 72, 76, 82, 88, 96, 99, 109, 115, 122, 130, 140, 144, 157, 164, 173, 183, 195, 201, 216, 225, 236, 248, 263, 270, 288, 299, 312, 327, 344, 353, 374

Offset: 1

Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009

Also the dimension of the largest Hecke-closed subspace of forms in S_k(Gamma_2) which satisfy the Ramanujan-Petersson conjecture. These forms are also characterized by the property that their (Andrianov) spinor zeta function does not have any pole.

			a(20)=1 because there is exactly one Siegel modular form of genus 2 and weight 20 which is not a lift of some form of genus 1.

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhaeusser, 1985.
T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256(3), p. 323-340, 1981.
R. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)

Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1).

Cf. A165684 for the full space of Siegel cusp forms. See also A029143, A027640, A165685.

For k > 1 we have a(k) = A165684(k) - A008615(2k-5).

Conjectured G.f.: -x^10*(x^7+x^6-x^2-x-1) / ((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)). - Colin Barker, Mar 30 2013

A319608 Irregular triangle read by rows: T(n,k) is the number of irreducible numerical semigroups with Frobenius number n and k minimal generators less than n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 5, 2, 1, 4, 1, 1, 4, 2, 1, 4, 2, 1, 7, 6, 1, 1, 4, 2, 1, 8, 9, 2, 1, 5, 4, 1, 1, 7, 8, 2, 1, 8, 9, 2, 1, 10, 17, 7, 1, 1, 5, 6, 2, 1, 10, 19, 12, 2, 1, 10, 16, 7, 1, 1, 10, 21, 11, 2, 1, 9, 16, 9, 2, 1, 13, 34, 26, 8, 1, 1, 8, 15, 10, 2, 1, 14, 41, 37, 14, 2

Offset: 1

Christopher O'Neill, Sep 24 2018

The length of each row is floor((n+1)/2) - floor(n/3).
Summing rows yields A158206.
The expected number of minimal generators of a randomly selected numerical semigroup S(M,p) equals Sum_{n=1..M} ( p * (1 - p)^(floor(n/2)) * Product_{k>=0} T(n,k)*p^k ).

			T(13,2) = 2, since {5,6,9} and {7,8,9,10,11,12} minimally generate irreducible numerical semigroups with Frobenius number 13.
When written in rows:
  1
  1
  1
  1
  1,  1
  1
  1,  2
  1,  1
  1,  2
  1,  2
  1,  4,  1
  1,  1
  1,  5,  2
  1,  4,  1
  1,  4,  2
  1,  4,  2
  1,  7,  6,  1
  1,  4,  2
  1,  8,  9,  2
  1,  5,  4,  1
  1,  7,  8,  2
  1,  8,  9,  2
  1, 10, 17,  7,  1
  1,  5,  6,  2
  1, 10, 19, 12,  2
  1, 10, 16,  7,  1
  1, 10, 21, 11,  2
  1,  9, 16,  9,  2
  1, 13, 34, 26,  8,  1
  1,  8, 15, 10,  2

J. De Loera, C. O'Neill, and D. Wilbourne, Random numerical semigroups and a simplicial complex of irreducible semigroups, arXiv:1710.00979 [math.AC], 2017.
C. Leng and C. O'Neill, A sequence of quasipolynomials arising from random numerical semigroups, arXiv:1809.09915 [math.CO], 2018.
C. O'Neill, The first 90 rows formatted as a triangle

Cf. A008615, A158206.

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A083024 Molien series for action of SL(3,C) on ternary forms of degree 4.

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A165686 Dimension of the space of Siegel cusp forms of genus 2 and weight 2k which are not Saito-Kurokawa lifts of forms of genus 1.

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A319608 Irregular triangle read by rows: T(n,k) is the number of irreducible numerical semigroups with Frobenius number n and k minimal generators less than n/2.

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