A391533 → x ↦ x+1 → (θ+1)²

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Triangle begins:
  0:    9,    24,      9;
  1:   18,    84,     72,      16;
  2:   36,   240,    333,     160,      25;
  3:   72,   624,   1206,     912,     300,      36;
  4:  144,  1536,   3816,    3968,    2025,     504,      49;
  5:  288,  3648,  11088,   14720,   10250,    3924,     784,     64;
  6:  576,  8448,  30384,   49152,   43500,   22608,    6909,   1152,     81;
  7: 1152, 19200,  79776,  152320,  163800,  107856,   44590,  11328,   1620,   100;
  8: 2304, 43008, 202752,  446464,  565600,  451584,  235984,  80896,  17577,  2200,  121;
  9: 4608, 95232, 502272, 1253376, 1828800, 1717632, 1086624, 470016, 137538, 26100, 2904, 144;

The row polynomials satisfy (36+(54*n+78)*x+(9*n^2+60*n+72)*x^2+(6*n^2+26*n+28)*x^3+(n^2+4*n+4)*x^4)*P(n,x) = (72+(108*n+300)*x+(18*n^2+210*n+414)*x^2+(21*n^2+154*n+261)*x^3+(8*n^2+50*n+78)*x^4+(n^2+6*n+9)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 9*x^2.

P[0, x_] := 9 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72 + (108*n + 300)*x + (18*n^2 + 210*n + 414)*x^2 + (21*n^2 + 154*n + 261)*x^3 + (8*n^2 + 50*n + 78)*x^4 + (n^2 + 6*n + 9)*x^5)*P[n - 1, x])/(36 + (54*n + 78)*x + (9*n^2 + 60*n + 72)*x^2 + (6*n^2 + 26*n + 28)*x^3 + (n^2 + 4*n + 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 24*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (108*n + 300)*x + (18*n^2 + 210*n + 414)*x^2 + (21*n^2 + 154*n + 261)*x^3 + (8*n^2 + 50*n + 78)*x^4 + (n^2 + 6*n + 9)*x^5)*v[i - 1])/(36 + (54*n + 78)*x + (9*n^2 + 60*n + 72)*x^2 + (6*n^2 + 26*n + 28)*x^3 + (n^2 + 4*n + 4)*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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  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 basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    • Geometric basis map (elementary) polytool ? step ›

      • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
                • ↑ Liu–Wang theorem (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
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.