A100862 → x ↦ x+1 → θ+1

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Triangle begins:
  0:     1;
  1:     2,      2;
  2:     5,     10,      3;
  3:    14,     42,     27,      4;
  4:    43,    172,    171,     56,      5;
  5:   142,    710,    960,    500,    100,      6;
  6:   499,   2994,   5130,   3700,   1200,    162,     7;
  7:  1850,  12950,  26901,  24920,  11375,   2520,   245,    8;
  8:  7193,  57544, 140574, 159488,  94675,  29904,  4802,  352,   9;
  9: 29186, 262674, 738072, 992712, 728910, 302778, 69972, 8496, 486, 10;

The row polynomials satisfy (12*x+(6*n+12)*x^2+(n^2+3*n-2)*x^3-2*n*x^4)*P(n,x) = -(8*n-8+(6*n^2+12*n-42)*x+(n^3+8*n^2-17*n-52)*x^2+(n^3-2*n^2-25*n-14)*x^3-(3*n^2+n-4)*x^4+(2*n+2)*x^5)*P(n-1,x) + (8+(8*n+28)*x+(2*n^2+22*n+28)*x^2+(4*n^2+16*n)*x^3+(2*n^2-2*n-12)*x^4-(4*n+4)*x^5)*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[(-(8*n - 8 + (6*n^2 + 12*n - 42)*x + (n^3 + 8*n^2 - 17*n - 52)*x^2 + (n^3 - 2*n^2 - 25*n - 14)*x^3 - (3*n^2 + n - 4)*x^4 + (2*n + 2)*x^5)*P[n - 1, x] + (8 + (8*n + 28)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n^2 + 16*n)*x^3 + (2*n^2 - 2*n - 12)*x^4 - (4*n + 4)*x^5)*D[P[n - 1, x], x])/(12*x + (6*n + 12)*x^2 + (n^2 + 3*n - 2)*x^3 - 2*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] = (-(8*n - 8 + (6*n^2 + 12*n - 42)*x + (n^3 + 8*n^2 - 17*n - 52)*x^2 + (n^3 - 2*n^2 - 25*n - 14)*x^3 - (3*n^2 + n - 4)*x^4 + (2*n + 2)*x^5)*v[i - 1] + (8 + (8*n + 28)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n^2 + 16*n)*x^3 + (2*n^2 - 2*n - 12)*x^4 - (4*n + 4)*x^5)*deriv(v[i - 1], x))/(12*x + (6*n + 12)*x^2 + (n^2 + 3*n - 2)*x^3 - 2*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 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete gra…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
        • 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 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
            • ↑ 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 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
        • ↑ 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) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
        • ↑ 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) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
        • ↑ 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) (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.