A391533 → reverse → θ+1

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Submission text ?

Triangle begins:
  0: 1,  8,  12;
  1: 1, 10,  24,   16;
  2: 1, 12,  39,   48,   20;
  3: 1, 14,  57,  100,   80,   24;
  4: 1, 16,  78,  176,  205,  120,   28;
  5: 1, 18, 102,  280,  425,  366,  168,   32;
  6: 1, 20, 129,  416,  775,  876,  595,  224,   36;
  7: 1, 22, 159,  588, 1295, 1806, 1617,  904,  288,   40;
  8: 1, 24, 192,  800, 2030, 3360, 3724, 2752, 1305,  360,  44;
  9: 1, 26, 228, 1056, 3030, 5796, 7644, 7008, 4401, 1810, 440, 48;

The row polynomials satisfy (1+(n+6)*x+(2*n+4)*x^2)*P(n,x) = (1+(n+8)*x+(3*n+13)*x^2+(2*n+6)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 12*x^2.

P[0, x_] := 1 + 8*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (n + 8)*x + (3*n + 13)*x^2 + (2*n + 6)*x^3)*P[n - 1, x])/(1 + (n + 6)*x + (2*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 8*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 8)*x + (3*n + 13)*x^2 + (2*n + 6)*x^3)*v[i - 1])/(1 + (n + 6)*x + (2*n + 4)*x^2)); 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 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 [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.?
            no recorded property covers this requirement
        • 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

Related ?

Same fingerprint ?

• A159854 → basis A013609 → θ+1

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.