A028338 → basis A008277 → (θ+1)²

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Triangle begins:
  0:         1;
  1:         2,          4;
  2:         8,         28,          9;
  3:        48,        228,        135,         16;
  4:       384,       2244,       1863,        416,        25;
  5:      3840,      26220,      27405,       8720,      1000,       36;
  6:     46080,     356220,     442665,     179520,     29625,     2052,      49;
  7:    645120,    5527620,    7884135,    3838800,    813750,    81648,    3773,     64;
  8:  10321920,   96550020,  154351575,   86953440,  22273125,  2886408,  194138,   6400,    81;
  9: 185794560, 1875738060, 3304269045, 2100425040, 625873500, 98460684, 8588034, 413568, 10206, 100;

The row polynomials satisfy (4-8*x+(28*n-4)*x^2-(18*n-4)*x^3+4*n^2*x^4)*P(n,x) = (8*n-(16*n-16)*x+(56*n^2-8*n-32)*x^2-(36*n^2-62*n-42)*x^3+(8*n^3-32*n-24)*x^4+(4*n^2+8*n+4)*x^5)*P(n-1,x) + (4*x-15*x^2+(28*n+39)*x^3-(28*n+29)*x^4+(4*n^2+12*n+6)*x^5)*d/dx P(n-1,x) + (9*x^3-9*x^4+(4*n+2)*x^5)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - (16*n - 16)*x + (56*n^2 - 8*n - 32)*x^2 - (36*n^2 - 62*n - 42)*x^3 + (8*n^3 - 32*n - 24)*x^4 + (4*n^2 + 8*n + 4)*x^5)*P[n - 1, x] + (4*x - 15*x^2 + (28*n + 39)*x^3 - (28*n + 29)*x^4 + (4*n^2 + 12*n + 6)*x^5)*D[P[n - 1, x], x] + (9*x^3 - 9*x^4 + (4*n + 2)*x^5)*D[P[n - 1, x], {x, 2}])/(4 - 8*x + (28*n - 4)*x^2 - (18*n - 4)*x^3 + 4*n^2*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] = ((8*n - (16*n - 16)*x + (56*n^2 - 8*n - 32)*x^2 - (36*n^2 - 62*n - 42)*x^3 + (8*n^3 - 32*n - 24)*x^4 + (4*n^2 + 8*n + 4)*x^5)*v[i - 1] + (4*x - 15*x^2 + (28*n + 39)*x^3 - (28*n + 29)*x^4 + (4*n^2 + 12*n + 6)*x^5)*deriv(v[i - 1], x) + (9*x^3 - 9*x^4 + (4*n + 2)*x^5)*deriv(deriv(v[i - 1], x), x))/(4 - 8*x + (28*n - 4)*x^2 - (18*n - 4)*x^3 + 4*n^2*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_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (above)
                • 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
            • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.