A011973 → D → (θ+1)²

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Triangle begins:
  0: 0;
  1: 0;
  2: 1;
  3: 2;
  4: 3,   8;
  5: 4,  24;
  6: 5,  48,  27;
  7: 6,  80, 108;
  8: 7, 120, 270,  64;
  9: 8, 168, 540, 320;

The row polynomials satisfy (n^3-2*n^2+(3*n^4-18*n^3+33*n^2-16*n-4)*x+(n^5-8*n^4+25*n^3-38*n^2+28*n-8)*x^2)*P(n,x) = (n^3-n^2+(3*n^4-7*n^3-3*n^2-7*n-2)*x+(n^5+3*n^4-33*n^3+45*n^2+4*n-4)*x^2+(2*n^5-10*n^4+14*n^3-6*n^2)*x^3)*P(n-1,x) - (n^2*x+(3*n^3-2*n^2+n)*x^2+(n^4+10*n^3-23*n^2+4*n)*x^3+(4*n^4-8*n^3+4*n^2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 + (3*n^4 - 7*n^3 - 3*n^2 - 7*n - 2)*x + (n^5 + 3*n^4 - 33*n^3 + 45*n^2 + 4*n - 4)*x^2 + (2*n^5 - 10*n^4 + 14*n^3 - 6*n^2)*x^3)*P[n - 1, x] - (n^2*x + (3*n^3 - 2*n^2 + n)*x^2 + (n^4 + 10*n^3 - 23*n^2 + 4*n)*x^3 + (4*n^4 - 8*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + (3*n^4 - 18*n^3 + 33*n^2 - 16*n - 4)*x + (n^5 - 8*n^4 + 25*n^3 - 38*n^2 + 28*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - n^2 + (3*n^4 - 7*n^3 - 3*n^2 - 7*n - 2)*x + (n^5 + 3*n^4 - 33*n^3 + 45*n^2 + 4*n - 4)*x^2 + (2*n^5 - 10*n^4 + 14*n^3 - 6*n^2)*x^3)*v[i - 1] - (n^2*x + (3*n^3 - 2*n^2 + n)*x^2 + (n^4 + 10*n^3 - 23*n^2 + 4*n)*x^3 + (4*n^4 - 8*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + (3*n^4 - 18*n^3 + 33*n^2 - 16*n - 4)*x + (n^5 - 8*n^4 + 25*n^3 - 38*n^2 + 28*n - 8)*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 derivative of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficient…?
    • Laguerre’s theorem on differential operators [SP14] 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
    • Laguerre’s theorem on differential operators [SP14] 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 → D → (θ+1)²
• A169803 → D → (θ+1)²

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.