A161198 → (θ+1)²

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Submission text ?

Triangle begins:
   1:        1;
   2:        1,         8;
   3:        3,        32,         36;
   4:       15,       184,        324,        128;
   5:      105,      1408,       3096,       2048,        400;
   6:      945,     13512,      34200,      29440,      10000,      1152;
   7:    10395,    156192,     437004,     445440,     202000,     41472,      3136;
   8:   135135,   2113656,    6383916,    7344512,    4018000,   1120896,    153664,     8192;
   9:  2027025,  32785920,  105270192,  132866048,   83221600,  28385280,   5356288,   524288,   20736;
  10: 34459425, 573576840, 1937129904, 2633016832, 1829973600, 722227968, 168327936, 22904832, 1679616, 51200;

The row polynomials satisfy P(n,x) = -(4*n^2-20*n+25+(16*n-40)*x-36*x^2)*P(n-2,x) + ((16*n-40)*x^2+16*x^3)*d/dx P(n-2,x) + (4*n-8+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.

P[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 20*n + 25 + (16*n - 40)*x - 36*x^2)*P[n - 2, x] + ((16*n - 40)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (4*n - 8 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 20*n + 25 + (16*n - 40)*x - 36*x^2)*v[i - 2] + ((16*n - 40)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (4*n - 8 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  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 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Parent-row provers of polytool polytool ? step › direct check
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A204420 → basis A013609 → (θ+1)²
• A290319 → basis A013611 → (θ+1)²

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.