A011973 → basis A013610 → θ+1

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Triangle begins:
  0:  1;
  1:  1;
  2:  2,   6;
  3:  3,  12;
  4:  5,  30,   27;
  5:  8,  60,   81;
  6: 13, 120,  243,  108;
  7: 21, 228,  594,  432;
  8: 34, 426, 1377, 1512,  405;
  9: 55, 780, 2997, 4320, 2025;

The row polynomials satisfy (30*n+(45*n^2+63*n)*x+(27*n^3+27*n^2)*x^2)*P(n,x) = (90*n-60+(180*n^2+324*n-396)*x+(81*n^3+567*n^2+162*n-756)*x^2+(162*n^3+324*n^2-162*n-324)*x^3)*P(n-1,x) - (25+(75*n+240)*x+(45*n^2+540*n+837)*x^2+(243*n^2+1269*n+1242)*x^3+(324*n^2+972*n+648)*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[((90*n - 60 + (180*n^2 + 324*n - 396)*x + (81*n^3 + 567*n^2 + 162*n - 756)*x^2 + (162*n^3 + 324*n^2 - 162*n - 324)*x^3)*P[n - 1, x] - (25 + (75*n + 240)*x + (45*n^2 + 540*n + 837)*x^2 + (243*n^2 + 1269*n + 1242)*x^3 + (324*n^2 + 972*n + 648)*x^4)*D[P[n - 1, x], x])/(30*n + (45*n^2 + 63*n)*x + (27*n^3 + 27*n^2)*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] = ((90*n - 60 + (180*n^2 + 324*n - 396)*x + (81*n^3 + 567*n^2 + 162*n - 756)*x^2 + (162*n^3 + 324*n^2 - 162*n - 324)*x^3)*v[i - 1] - (25 + (75*n + 240)*x + (45*n^2 + 540*n + 837)*x^2 + (243*n^2 + 1269*n + 1242)*x^3 + (324*n^2 + 972*n + 648)*x^4)*deriv(v[i - 1], x))/(30*n + (45*n^2 + 63*n)*x + (27*n^3 + 27*n^2)*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 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 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 ?

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

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.