A114656 → basis A013609 → D+1

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Triangle begins:
   1:       1;
   2:       5,        2;
   3:      27,       24,        4;
   4:     147,      222,       84,        8;
   5:     805,     1776,     1160,      256,       16;
   6:    4433,    13050,    12680,     4880,      720,       32;
   7:   24535,    90936,   119820,    69440,    18000,     1920,      64;
   8:  136391,   611702,  1028412,   817880,   322000,    60704,    4928,    128;
   9:  761097,  4014240,  8255632,  8465408,  4614624,  1329664,  191744,  12288,   256;
  10: 4261149, 25870770, 63127440, 79890720, 56619360, 22744512, 5037312, 576000, 29952, 512;

The row polynomials satisfy (4*n^3-4*n^2-n+7+(8*n^2-8*n-16)*x+(4*n+4)*x^2)*P(n,x) = (20*n^3-20*n^2+13*n-1+(24*n^3-16*n^2-46*n+6)*x+(48*n^2-100*n+68)*x^2+(24*n-24)*x^3)*P(n-1,x) + (4*n^2-2*n+1+(8*n-8)*x-(16*n^2-8*n)*x^2-(32*n-32)*x^3-16*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[((20*n^3 - 20*n^2 + 13*n - 1 + (24*n^3 - 16*n^2 - 46*n + 6)*x + (48*n^2 - 100*n + 68)*x^2 + (24*n - 24)*x^3)*P[n - 1, x] + (4*n^2 - 2*n + 1 + (8*n - 8)*x - (16*n^2 - 8*n)*x^2 - (32*n - 32)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(4*n^3 - 4*n^2 - n + 7 + (8*n^2 - 8*n - 16)*x + (4*n + 4)*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] = ((20*n^3 - 20*n^2 + 13*n - 1 + (24*n^3 - 16*n^2 - 46*n + 6)*x + (48*n^2 - 100*n + 68)*x^2 + (24*n - 24)*x^3)*v[i - 1] + (4*n^2 - 2*n + 1 + (8*n - 8)*x - (16*n^2 - 8*n)*x^2 - (32*n - 32)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(4*n^3 - 4*n^2 - n + 7 + (8*n^2 - 8*n - 16)*x + (4*n + 4)*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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_A013609 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