A114656 → basis A013610 → θ+1

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Triangle begins:
   1:      1;
   2:      3,        6;
   3:     11,       48,       27;
   4:     45,      306,      405,       108;
   5:    197,     1824,     3942,      2592,       405;
   6:    903,    10590,    32130,     35640,     14175,      1458;
   7:   4279,    60768,   238545,    380160,    261225,     69984,      5103;
   8:  20793,   346794,  1674351,   3511620,   3529575,   1663578,    321489,    17496;
   9: 103049,  1974144, 11327148,  29598912,  39695670,  27760320,   9573228,  1399680,   59049;
  10: 518859, 11226870, 74683620, 234284400, 395737650, 373478364, 193322052, 51018336, 5845851, 196830;

The row polynomials satisfy -(6*n+6-(18*n^2+18*n)*x-(27*n^3-27*n)*x^2)*P(n,x) = -(30*n-6-(72*n^2-72*n+72)*x-(135*n^3+162*n^2-297*n)*x^2-(243*n^3-162*n^2-243*n+162)*x^3)*P(n-1,x) - (2-12*n*x-(18*n^2+9*n+9)*x^2+108*n*x^3+(162*n^2+81*n-81)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 6 - (72*n^2 - 72*n + 72)*x - (135*n^3 + 162*n^2 - 297*n)*x^2 - (243*n^3 - 162*n^2 - 243*n + 162)*x^3)*P[n - 1, x] - (2 - 12*n*x - (18*n^2 + 9*n + 9)*x^2 + 108*n*x^3 + (162*n^2 + 81*n - 81)*x^4)*D[P[n - 1, x], x])/(-6*n - 6 + (18*n^2 + 18*n)*x + (27*n^3 - 27*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(30*n - 6 - (72*n^2 - 72*n + 72)*x - (135*n^3 + 162*n^2 - 297*n)*x^2 - (243*n^3 - 162*n^2 - 243*n + 162)*x^3)*v[i - 1] - (2 - 12*n*x - (18*n^2 + 9*n + 9)*x^2 + 108*n*x^3 + (162*n^2 + 81*n - 81)*x^4)*deriv(v[i - 1], x))/(-6*n - 6 + (18*n^2 + 18*n)*x + (27*n^3 - 27*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  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 basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two c…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Three-term recurrence criterion (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (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