A122848 → basis A013610 → θ+1

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Triangle begins:
  0:    1;
  1:    1,     6;
  2:    2,    18,      27;
  3:    4,    54,     162,     108;
  4:   10,   168,     729,    1080,      405;
  5:   26,   540,    3105,    7020,     6075,     1458;
  6:   76,  1836,   12960,   39420,    54675,    30618,     5103;
  7:  232,  6468,   54432,  207900,   396900,   367416,   142884,    17496;
  8:  764, 23904,  232092, 1064448,  2594025,  3388392,  2214702,   629856,   59049;
  9: 2620, 91368, 1007964, 5397840, 15997905, 27096930, 25504794, 12282192, 2657205, 196830;

The row polynomials satisfy (24+(36*n+72)*x+(27*n^2+27*n)*x^2-162*n*x^3)*P(n,x) = -(24*n-48+(54*n^2-54*n-252)*x+(27*n^3-27*n^2-378*n-324)*x^2-(243*n^2+81*n-162)*x^3+(486*n+486)*x^4)*P(n-1,x) + (8+(24*n+48)*x+(18*n^2+90*n+36)*x^2+(54*n^2-54*n-216)*x^3-(324*n+324)*x^4)*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[(-(24*n - 48 + (54*n^2 - 54*n - 252)*x + (27*n^3 - 27*n^2 - 378*n - 324)*x^2 - (243*n^2 + 81*n - 162)*x^3 + (486*n + 486)*x^4)*P[n - 1, x] + (8 + (24*n + 48)*x + (18*n^2 + 90*n + 36)*x^2 + (54*n^2 - 54*n - 216)*x^3 - (324*n + 324)*x^4)*D[P[n - 1, x], x])/(24 + (36*n + 72)*x + (27*n^2 + 27*n)*x^2 - 162*n*x^3)]];
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] = (-(24*n - 48 + (54*n^2 - 54*n - 252)*x + (27*n^3 - 27*n^2 - 378*n - 324)*x^2 - (243*n^2 + 81*n - 162)*x^3 + (486*n + 486)*x^4)*v[i - 1] + (8 + (24*n + 48)*x + (18*n^2 + 90*n + 36)*x^2 + (54*n^2 - 54*n - 216)*x^3 - (324*n + 324)*x^4)*deriv(v[i - 1], x))/(24 + (36*n + 72)*x + (27*n^2 + 27*n)*x^2 - 162*n*x^3)); 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_A013610 of Exponential Riordan array (1, x(1+x/2)).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Exponential Riordan array (1, x(1+x/2)).?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).?
            • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? 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)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Exponential Riordan array (1, x(1+x/2)).?
                • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Exponential Riordan array (1, x(1+x/2)).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Parent-row provers of polytool polytool ? step › direct check
            • 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 for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Exponential Riordan array (1, x(1+x/2)).?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Exponential Riordan array (1, x(1+x/2)).?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (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.