A128813 → basis A013610 → x ↦ x/(x+1)

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Triangle begins:
  0:           1;
  1:           2,            5;
  2:           8,           46,            65;
  3:          56,          522,          1605,           1625;
  4:         616,         8038,         39057,          83680,           66625;
  5:        9856,       166184,       1115230,        3721357,         6170480,         4064125;
  6:      216832,      4493808,      38660700,      176664404,       452065905,       613901550,        345450625;
  7:     6288128,    154822448,    1628960604,     9491926816,     33072988897,     68886592215,      79388943275,      39035920625;
  8:   232660736,   6640209136,   82720797308,   587400579772,   2600029977509,   7344387302020,   12925946772350,   12955725838000,    5660208490625;
  9: 10702393856, 347561213472, 5007034529784, 41992890982260, 225920883904146, 808447241822049, 1923927653193720, 2935559754343350, 2605355967246750, 1024497736803125;

The row polynomials satisfy (2+2*x)*P(n,x) = (n^2+n+2+(5*n^2+5*n+4)*x+(4*n^2+4*n+2)*x^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[((n^2 + n + 2 + (5*n^2 + 5*n + 4)*x + (4*n^2 + 4*n + 2)*x^2)*P[n - 1, x])/(2 + 2*x)]];
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] = ((n^2 + n + 2 + (5*n^2 + 5*n + 4)*x + (4*n^2 + 4*n + 2)*x^2)*v[i - 1])/(2 + 2*x)); 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 [-1, 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_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (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.