A011973 → basis A013610 → D+1

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Triangle begins:
  0:   1;
  1:   1;
  2:   5,    3;
  3:   9,    6;
  4:  20,   33,    9;
  5:  38,   84,   27;
  6:  73,  222,  162,   27;
  7: 135,  510,  522,  108;
  8: 247, 1131, 1593,  702,  81;
  9: 445, 2388, 4239, 2700, 405;

The row polynomials satisfy (27*n^3-36*n^2-30*n+(108*n^2-189*n)*x+108*n*x^2)*P(n,x) = (81*n^3-135*n^2-63*n+78+(162*n^3-54*n^2-585*n+396)*x+(648*n^2-810*n+270)*x^2+(648*n-648)*x^3)*P(n-1,x) - (45*n^2+30*n-5+(243*n^2+342*n+18)*x+(324*n^2+1188*n+387)*x^2+(1296*n+1296)*x^3+1296*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[((81*n^3 - 135*n^2 - 63*n + 78 + (162*n^3 - 54*n^2 - 585*n + 396)*x + (648*n^2 - 810*n + 270)*x^2 + (648*n - 648)*x^3)*P[n - 1, x] - (45*n^2 + 30*n - 5 + (243*n^2 + 342*n + 18)*x + (324*n^2 + 1188*n + 387)*x^2 + (1296*n + 1296)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(27*n^3 - 36*n^2 - 30*n + (108*n^2 - 189*n)*x + 108*n*x^2)]];
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] = ((81*n^3 - 135*n^2 - 63*n + 78 + (162*n^3 - 54*n^2 - 585*n + 396)*x + (648*n^2 - 810*n + 270)*x^2 + (648*n - 648)*x^3)*v[i - 1] - (45*n^2 + 30*n - 5 + (243*n^2 + 342*n + 18)*x + (324*n^2 + 1188*n + 387)*x^2 + (1296*n + 1296)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(27*n^3 - 36*n^2 - 30*n + (108*n^2 - 189*n)*x + 108*n*x^2)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coef…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
        • Parent-row provers of polytool polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

          • from delannoy?
            • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from delannoy?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from delannoy?
                    • ↑ Liu–Wang theorem with a lag row (above)
                    • ↑ Liu–Wang theorem for three-term recurrences (above)
                    • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                      • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                        no recorded property covers this requirement
                    • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                      • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                        no recorded property covers this requirement
                • ↑ Affine-current basis map theorem (above)
                • ↑ Affine-current basis map theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine-current basis map theorem (above)
            • ↑ Affine-current basis map theorem (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
    • Geometric basis map (elementary) polytool ? step ›

      • from Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Gamma criterion (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A052553 → basis A013610 → D+1
• A169803 → basis A013610 → D+1

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.