A367270 → x ↦ x+1 → D+1

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Triangle begins:
  0:      1;
  1:      1;
  2:      5,      2;
  3:     22,     18,      3;
  4:     95,    120,     42,      4;
  5:    406,    700,    390,     80,      5;
  6:   1722,   3780,   2940,    980,    135,     6;
  7:   7260,  19404,  19530,   9240,   2100,   210,    7;
  8:  30459,  96096, 119196,  73920,  24150,  4032,  308,   8;
  9: 127270, 463320, 684684, 528528, 228690, 55440, 7140, 432, 9;

The row polynomials satisfy (n^4-2*n^3+n+(2*n^3-5*n^2+3*n)*x+(n^2-n)*x^2)*P(n,x) = (4*n^4-6*n^3-8*n^2+16*n-6+(3*n^4+2*n^3-25*n^2+30*n-10)*x+(6*n^3-12*n^2+8*n-2)*x^2+(3*n^2-5*n+2)*x^3)*P(n-1,x) - ((2*n^3-4*n+2)*x+(2*n^3+4*n^2-8*n+3)*x^2+(4*n^2-2*n)*x^3+(2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 - 6*n^3 - 8*n^2 + 16*n - 6 + (3*n^4 + 2*n^3 - 25*n^2 + 30*n - 10)*x + (6*n^3 - 12*n^2 + 8*n - 2)*x^2 + (3*n^2 - 5*n + 2)*x^3)*P[n - 1, x] - ((2*n^3 - 4*n + 2)*x + (2*n^3 + 4*n^2 - 8*n + 3)*x^2 + (4*n^2 - 2*n)*x^3 + (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 2*n^3 + n + (2*n^3 - 5*n^2 + 3*n)*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^4 - 6*n^3 - 8*n^2 + 16*n - 6 + (3*n^4 + 2*n^3 - 25*n^2 + 30*n - 10)*x + (6*n^3 - 12*n^2 + 8*n - 2)*x^2 + (3*n^2 - 5*n + 2)*x^3)*v[i - 1] - ((2*n^3 - 4*n + 2)*x + (2*n^3 + 4*n^2 - 8*n + 3)*x^2 + (4*n^2 - 2*n)*x^3 + (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 2*n^3 + n + (2*n^3 - 5*n^2 + 3*n)*x + (n^2 - n)*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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from basis_map_A007318 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
        • 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
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

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