A069466 → γ → D

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Triangle begins:
  0:       0;
  1:       0;
  2:      12;
  3:     120;
  4:     840,      840;
  5:    5040,    15120;
  6:   27720,   166320,     55440;
  7:  144144,  1441440,   1441440;
  8:  720720, 10810800,  21621600,   3603600;
  9: 3500640, 73513440, 245044800, 122522400;

The row polynomials satisfy (n^2-2*n)*P(n,x) = (4*n^2-2*n+(16*n^2-56*n+24)*x)*P(n-1,x) + ((8*n-4)*x-(32*n-16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 12.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 12;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (16*n^2 - 56*n + 24)*x)*P[n - 1, x] + ((8*n - 4)*x - (32*n - 16)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 12; for(n = 3, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n + (16*n^2 - 56*n + 24)*x)*v[i - 1] + ((8*n - 4)*x - (32*n - 16)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n)); 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 ?

• A132812 → γ
• A391403 → D
• A089627 → D
• A107131: A Motzkin related triangle.
• A055151: Triangular array of Motzkin polynomial coefficients.
• narayana → γ

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