A062145 → x ↦ x+1 → θ+1

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Triangle begins:
   1:      1;
   2:      5,       8;
   3:     21,      60,      30;
   4:     84,     336,     315,       80;
   5:    330,    1680,    2268,     1120,      175;
   6:   1287,    7920,   13860,    10080,     3150,     336;
   7:   5005,   36036,   77220,    73920,    34650,    7560,     588;
   8:  19448,  160160,  405405,   480480,   300300,   99792,   16170,    960;
   9:  75582,  700128, 2042040,  2882880,  2252250, 1009008,  252252,  31680,  1485;
  10: 293930, 3023280, 9976824, 16336320, 15315300, 8648640, 2942940, 576576, 57915, 2200;

The row polynomials satisfy (2*n^3+3*n^2-3*n-2+(n^4+2*n^3-3*n^2-4*n+4)*x)*P(n,x) = (8*n^3+8*n^2+2*n+(4*n^4+18*n^3+4*n^2+6*n+4)*x+(3*n^4+8*n^3+9*n-2)*x^2)*P(n-1,x) - ((4*n^2+4*n+1)*x+(2*n^3+10*n^2+4*n+2)*x^2+(2*n^3+6*n^2+1)*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[((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 18*n^3 + 4*n^2 + 6*n + 4)*x + (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*P[n - 1, x] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 10*n^2 + 4*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*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] = ((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 18*n^3 + 4*n^2 + 6*n + 4)*x + (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*v[i - 1] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 10*n^2 + 4*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from shift_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochha…?
    • 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) 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) 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.