A391533 → basis A013609 → reverse

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Submission text ?

Triangle begins:
  0:    4,    12,      9;
  1:    8,    32,     42,     18;
  2:   16,    80,    148,    120,     36;
  3:   32,   192,    456,    536,    312,      72;
  4:   64,   448,   1296,   1984,   1696,     768,     144;
  5:  128,  1024,   3488,   6560,   7360,    4928,    1824,     288;
  6:  256,  2304,   9024,  20096,  27840,   24576,   13504,    4224,    576;
  7:  512,  5120,  22656,  58240,  95872,  104832,   76160,   35456,   9600,   1152;
  8: 1024, 11264,  55552, 161792, 308224,  401408,  361984,  223232,  90112,  21504,  2304;
  9: 2048, 24576, 133632, 434688, 940032, 1419264, 1526784, 1170432, 626688, 223232, 47616, 4608;

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 12*x + 9*x^2.

P[0, x_] := 4 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 12*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1]); 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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013609 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)

Related ?

Same fingerprint ?

• A159854 → x ↦ x+1 → basis A013610
• A391533 → basis A013609

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.