A062145 → reverse → θ+1

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Triangle begins:
   1:   1;
   2:   4,    2;
   3:  10,   20,     3;
   4:  20,   90,    54,      4;
   5:  35,  280,   378,    112,      5;
   6:  56,  700,  1680,   1120,    200,      6;
   7:  84, 1512,  5670,   6720,   2700,    324,      7;
   8: 120, 2940, 15876,  29400,  21000,   5670,    490,     8;
   9: 165, 5280, 38808, 103488, 115500,  55440,  10780,   704,   9;
  10: 220, 8910, 85536, 310464, 498960, 374220, 129360, 19008, 972, 10;

The row polynomials satisfy (2*n^2+2*n-4+(n^4-n^3-3*n^2+5*n-2)*x)*P(n,x) = (2*n^2+8*n+8+(n^4+2*n^3+9*n^2-12*n-12)*x+(3*n^4-4*n^3-3*n^2+4)*x^2)*P(n-1,x) + ((4*n+2)*x+(2*n^3-4*n-4)*x^2-(2*n^3-2)*x^3)*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[((2*n^2 + 8*n + 8 + (n^4 + 2*n^3 + 9*n^2 - 12*n - 12)*x + (3*n^4 - 4*n^3 - 3*n^2 + 4)*x^2)*P[n - 1, x] + ((4*n + 2)*x + (2*n^3 - 4*n - 4)*x^2 - (2*n^3 - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 2*n - 4 + (n^4 - n^3 - 3*n^2 + 5*n - 2)*x)]];
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] = ((2*n^2 + 8*n + 8 + (n^4 + 2*n^3 + 9*n^2 - 12*n - 12)*x + (3*n^4 - 4*n^3 - 3*n^2 + 4)*x^2)*v[i - 1] + ((4*n + 2)*x + (2*n^3 - 4*n - 4)*x^2 - (2*n^3 - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 2*n - 4 + (n^4 - n^3 - 3*n^2 + 5*n - 2)*x)); 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 reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n -…?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n -…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (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. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.