A069466 → θ+1

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Submission text ?

Triangle begins:
  0:     1;
  1:     2,       4;
  2:     6,      48,        18;
  3:    20,     360,       540,         80;
  4:    70,    2240,      7560,       4480,        350;
  5:   252,   12600,     75600,     100800,      31500,       1512;
  6:   924,   66528,    623700,    1478400,    1039500,     199584,       6468;
  7:  3432,  336336,   4540536,   16816800,   21021000,    9081072,    1177176,     27456;
  8: 12870, 1647360,  30270240,  161441280,  315315000,  242161920,   70630560,   6589440,   115830;
  9: 48620, 7876440, 189034560, 1372250880, 3859455600, 4631346720, 2401439040, 504092160, 35443980, 486200;

The row polynomials satisfy -(n^2-n^4*x)*P(n,x) = -(4*n^2-2*n-(4*n^4-2*n^3-24*n^2+28*n-8)*x-(12*n^4+2*n^3-16*n^2+6*n)*x^2)*P(n-1,x) - ((8*n-4)*x-(8*n^3+8*n^2+2*n-4)*x^2+(8*n^3+8*n^2-6*n)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 2*n - (4*n^4 - 2*n^3 - 24*n^2 + 28*n - 8)*x - (12*n^4 + 2*n^3 - 16*n^2 + 6*n)*x^2)*P[n - 1, x] - ((8*n - 4)*x - (8*n^3 + 8*n^2 + 2*n - 4)*x^2 + (8*n^3 + 8*n^2 - 6*n)*x^3)*D[P[n - 1, x], x])/(-n^2 + n^4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^2 - 2*n - (4*n^4 - 2*n^3 - 24*n^2 + 28*n - 8)*x - (12*n^4 + 2*n^3 - 16*n^2 + 6*n)*x^2)*v[i - 1] - ((8*n - 4)*x - (8*n^3 + 8*n^2 + 2*n - 4)*x^2 + (8*n^3 + 8*n^2 - 6*n)*x^3)*deriv(v[i - 1], x))/(-n^2 + n^4*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A008459 → D → (θ+1)²
• A008459 → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023