A114687 → θ+1 → D+1

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Triangle begins:
  0:   1;
  1:   5,     4;
  2:  13,    36,     12;
  3:  25,   168,    168,      32;
  4:  41,   520,   1200,     640,      80;
  5:  61,  1260,   5400,    6400,    2160,     192;
  6:  85,  2604,  18060,   39200,   28560,    6720,     448;
  7: 113,  4816,  49392,  172480,  227360,  112896,   19712,    1024;
  8: 145,  8208, 116928,  602112, 1270080, 1128960,  408576,   55296,   2304;
  9: 181, 13140, 248400, 1774080, 5503680, 7789824, 4999680, 1382400, 149760, 5120;

The row polynomials satisfy (2*n^4-n^2+n+(4*n^3-n^2-5*n)*x+(2*n^2+2*n)*x^2)*P(n,x) = (2*n^4+4*n^3+3*n^2+n+(12*n^4+4*n^3-13*n^2-5*n)*x+(24*n^3-8*n^2-4*n)*x^2+(12*n^2+4*n)*x^3)*P(n-1,x) + ((4*n^3+4*n^2)*x-(8*n^3-2*n+1)*x^2-(16*n^2-4)*x^3-(8*n+4)*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[((2*n^4 + 4*n^3 + 3*n^2 + n + (12*n^4 + 4*n^3 - 13*n^2 - 5*n)*x + (24*n^3 - 8*n^2 - 4*n)*x^2 + (12*n^2 + 4*n)*x^3)*P[n - 1, x] + ((4*n^3 + 4*n^2)*x - (8*n^3 - 2*n + 1)*x^2 - (16*n^2 - 4)*x^3 - (8*n + 4)*x^4)*D[P[n - 1, x], x])/(2*n^4 - n^2 + n + (4*n^3 - n^2 - 5*n)*x + (2*n^2 + 2*n)*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] = ((2*n^4 + 4*n^3 + 3*n^2 + n + (12*n^4 + 4*n^3 - 13*n^2 - 5*n)*x + (24*n^3 - 8*n^2 - 4*n)*x^2 + (12*n^2 + 4*n)*x^3)*v[i - 1] + ((4*n^3 + 4*n^2)*x - (8*n^3 - 2*n + 1)*x^2 - (16*n^2 - 4)*x^3 - (8*n + 4)*x^4)*deriv(v[i - 1], x))/(2*n^4 - n^2 + n + (4*n^3 - n^2 - 5*n)*x + (2*n^2 + 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 ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from theta_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 …?
        • 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
        • 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 …?
                • ↑ 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) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 …?
        • ↑ 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) (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