A069466 → γ → D+1

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Submission text ?

Triangle begins:
  0:       1;
  1:       2;
  2:      18,       12;
  3:     140,      120;
  4:     910,     1680,       420;
  5:    5292,    20160,      7560;
  6:   28644,   194040,    138600,     18480;
  7:  147576,  1585584,   2162160,    480480;
  8:  733590, 11531520,  27027000,  10810800,   900900;
  9: 3549260, 77014080, 281801520, 204204000, 30630600;

The row polynomials satisfy (n^4-3*n^3+3*n^2+(4*n^3-11*n^2)*x+4*n^2*x^2)*P(n,x) = (4*n^4-6*n^3+6*n^2-2*n+(16*n^4-56*n^3+28*n^2-2*n)*x+(64*n^3-192*n^2+192*n-56)*x^2+(64*n^2-96*n+32)*x^3)*P(n-1,x) + ((8*n^3-12*n^2+4*n)*x-(32*n^3-80*n^2+56*n-12)*x^2-(128*n^2-192*n+64)*x^3-(128*n-64)*x^4)*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^4 - 6*n^3 + 6*n^2 - 2*n + (16*n^4 - 56*n^3 + 28*n^2 - 2*n)*x + (64*n^3 - 192*n^2 + 192*n - 56)*x^2 + (64*n^2 - 96*n + 32)*x^3)*P[n - 1, x] + ((8*n^3 - 12*n^2 + 4*n)*x - (32*n^3 - 80*n^2 + 56*n - 12)*x^2 - (128*n^2 - 192*n + 64)*x^3 - (128*n - 64)*x^4)*D[P[n - 1, x], x])/(n^4 - 3*n^3 + 3*n^2 + (4*n^3 - 11*n^2)*x + 4*n^2*x^2)]];
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^4 - 6*n^3 + 6*n^2 - 2*n + (16*n^4 - 56*n^3 + 28*n^2 - 2*n)*x + (64*n^3 - 192*n^2 + 192*n - 56)*x^2 + (64*n^2 - 96*n + 32)*x^3)*v[i - 1] + ((8*n^3 - 12*n^2 + 4*n)*x - (32*n^3 - 80*n^2 + 56*n - 12)*x^2 - (128*n^2 - 192*n + 64)*x^3 - (128*n - 64)*x^4)*deriv(v[i - 1], x))/(n^4 - 3*n^3 + 3*n^2 + (4*n^3 - 11*n^2)*x + 4*n^2*x^2)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps an…?
    • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? 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 ?

• A089627 → D+1
• A391403 → D+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