A249252 → basis A007318 → θ+1

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Triangle begins:
  0:      1;
  1:      5,       8;
  2:     17,      48,      24;
  3:     61,     248,     240,       64;
  4:    217,    1152,    1656,      896,      160;
  5:    773,    5064,    9648,     7872,     2880,     384;
  6:   2753,   21456,   50904,    55552,    30880,    8448,     896;
  7:   9805,   88600,  251568,   343872,   256960,  107136,   23296,    2048;
  8:  34921,  358944, 1186680,  1949312,  1831200, 1029120,  341376,   61440,   4608;
  9: 124373, 1432936, 5405808, 10371008, 11740480, 8313216, 3721984, 1021952, 156672, 10240;

The row polynomials satisfy (34*n-102+(17*n^2+95*n-234)*x+(4*n^3+54*n^2+70*n-180)*x^2+(8*n^3+44*n^2-4*n-48)*x^3+(8*n^2-8*n)*x^4)*P(n,x) = -(238*n+102+(187*n^2+733*n+234)*x+(28*n^3+338*n^2+690*n+180)*x^2+(24*n^3+156*n^2+188*n+48)*x^3+(8*n^2-8*n)*x^4)*P(n-1,x) + (289+(289*n+1377)*x+(68*n^2+1037*n+2546)*x^2+(148*n^2+1364*n+2302)*x^3+(96*n^2+792*n+1044)*x^4+(16*n^2+192*n+216)*x^5+(16*n+16)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(238*n + 102 + (187*n^2 + 733*n + 234)*x + (28*n^3 + 338*n^2 + 690*n + 180)*x^2 + (24*n^3 + 156*n^2 + 188*n + 48)*x^3 + (8*n^2 - 8*n)*x^4)*P[n - 1, x] + (289 + (289*n + 1377)*x + (68*n^2 + 1037*n + 2546)*x^2 + (148*n^2 + 1364*n + 2302)*x^3 + (96*n^2 + 792*n + 1044)*x^4 + (16*n^2 + 192*n + 216)*x^5 + (16*n + 16)*x^6)*D[P[n - 1, x], x])/(34*n - 102 + (17*n^2 + 95*n - 234)*x + (4*n^3 + 54*n^2 + 70*n - 180)*x^2 + (8*n^3 + 44*n^2 - 4*n - 48)*x^3 + (8*n^2 - 8*n)*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] = (-(238*n + 102 + (187*n^2 + 733*n + 234)*x + (28*n^3 + 338*n^2 + 690*n + 180)*x^2 + (24*n^3 + 156*n^2 + 188*n + 48)*x^3 + (8*n^2 - 8*n)*x^4)*v[i - 1] + (289 + (289*n + 1377)*x + (68*n^2 + 1037*n + 2546)*x^2 + (148*n^2 + 1364*n + 2302)*x^3 + (96*n^2 + 792*n + 1044)*x^4 + (16*n^2 + 192*n + 216)*x^5 + (16*n + 16)*x^6)*deriv(v[i - 1], x))/(34*n - 102 + (17*n^2 + 95*n - 234)*x + (4*n^3 + 54*n^2 + 70*n - 180)*x^2 + (8*n^3 + 44*n^2 - 4*n - 48)*x^3 + (8*n^2 - 8*n)*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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.