A249250 → x ↦ x+1 → θ+1

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Triangle begins:
  0:     1;
  1:     4,      6;
  2:    10,     24,      9;
  3:    28,     96,     72,     12;
  4:    76,    336,    378,    144,     15;
  5:   208,   1128,   1692,   1008,    240,     18;
  6:   568,   3648,   6840,   5568,   2130,    360,    21;
  7:  1552,  11520,  25920,  26784,  14220,   3888,   504,   24;
  8:  4240,  35712,  93744, 117504,  80100,  30672,  6426,  672,  27;
  9: 11584, 109152, 327456, 482688, 402480, 199368, 58716, 9888, 864, 30;

The row polynomials satisfy -(48-(12*n-96)*x-(12*n^2+18*n-64)*x^2-(3*n^3+12*n^2+n-16)*x^3-(n^2-n)*x^4)*P(n,x) = -(144*n+48+(108*n^2+420*n+96)*x+(18*n^3+186*n^2+372*n+64)*x^2+(15*n^3+78*n^2+95*n+16)*x^3+(2*n^2-2*n)*x^4)*P(n-1,x) + (144+(144*n+648)*x+(36*n^2+468*n+1092)*x^2+(72*n^2+528*n+846)*x^3+(39*n^2+231*n+288)*x^4+(3*n^2+28*n+31)*x^5+(n+1)*x^6)*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[(-(144*n + 48 + (108*n^2 + 420*n + 96)*x + (18*n^3 + 186*n^2 + 372*n + 64)*x^2 + (15*n^3 + 78*n^2 + 95*n + 16)*x^3 + (2*n^2 - 2*n)*x^4)*P[n - 1, x] + (144 + (144*n + 648)*x + (36*n^2 + 468*n + 1092)*x^2 + (72*n^2 + 528*n + 846)*x^3 + (39*n^2 + 231*n + 288)*x^4 + (3*n^2 + 28*n + 31)*x^5 + (n + 1)*x^6)*D[P[n - 1, x], x])/(-48 + (12*n - 96)*x + (12*n^2 + 18*n - 64)*x^2 + (3*n^3 + 12*n^2 + n - 16)*x^3 + (n^2 - n)*x^4)]];
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] = (-(144*n + 48 + (108*n^2 + 420*n + 96)*x + (18*n^3 + 186*n^2 + 372*n + 64)*x^2 + (15*n^3 + 78*n^2 + 95*n + 16)*x^3 + (2*n^2 - 2*n)*x^4)*v[i - 1] + (144 + (144*n + 648)*x + (36*n^2 + 468*n + 1092)*x^2 + (72*n^2 + 528*n + 846)*x^3 + (39*n^2 + 231*n + 288)*x^4 + (3*n^2 + 28*n + 31)*x^5 + (n + 1)*x^6)*deriv(v[i - 1], x))/(-48 + (12*n - 96)*x + (12*n^2 + 18*n - 64)*x^2 + (3*n^3 + 12*n^2 + n - 16)*x^3 + (n^2 - n)*x^4)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem with a lag row (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. [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.