A026729 → θ+1 → (θ+1)²

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1: 1;
   2: 0, 8;
   3: 0, 8, 27;
   4: 0, 0, 54,  64;
   5: 0, 0, 27, 192, 125;
   6: 0, 0,  0, 192, 500,  216;
   7: 0, 0,  0,  64, 750, 1080,  343;
   8: 0, 0,  0,   0, 500, 2160, 2058,   512;
   9: 0, 0,  0,   0, 125, 2160, 5145,  3584,  729;
  10: 0, 0,  0,   0,   0, 1080, 6860, 10752, 5832, 1000;

The row polynomials satisfy (n^5+n^4-n^3-n^2-(3*n^4-9*n^2+4*n+2)*x+(n^3-3*n^2+3*n-1)*x^2)*P(n,x) = -(2*n^5-6*n^3-4*n^2-(10*n^4+14*n^3-2*n^2+10*n+4)*x+(11*n^3+18*n^2+9*n-2)*x^2-2*n^2*x^3)*P(n-1,x) + ((4*n^4+8*n^3+4*n^2)*x+(n^4-10*n^3-23*n^2-4*n)*x^2-(3*n^3+2*n^2+n)*x^3+n^2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^5 - 6*n^3 - 4*n^2 - (10*n^4 + 14*n^3 - 2*n^2 + 10*n + 4)*x + (11*n^3 + 18*n^2 + 9*n - 2)*x^2 - 2*n^2*x^3)*P[n - 1, x] + ((4*n^4 + 8*n^3 + 4*n^2)*x + (n^4 - 10*n^3 - 23*n^2 - 4*n)*x^2 - (3*n^3 + 2*n^2 + n)*x^3 + n^2*x^4)*D[P[n - 1, x], x])/(n^5 + n^4 - n^3 - n^2 - (3*n^4 - 9*n^2 + 4*n + 2)*x + (n^3 - 3*n^2 + 3*n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^5 - 6*n^3 - 4*n^2 - (10*n^4 + 14*n^3 - 2*n^2 + 10*n + 4)*x + (11*n^3 + 18*n^2 + 9*n - 2)*x^2 - 2*n^2*x^3)*v[i - 1] + ((4*n^4 + 8*n^3 + 4*n^2)*x + (n^4 - 10*n^3 - 23*n^2 - 4*n)*x^2 - (3*n^3 + 2*n^2 + n)*x^3 + n^2*x^4)*deriv(v[i - 1], x))/(n^5 + n^4 - n^3 - n^2 - (3*n^4 - 9*n^2 + 4*n + 2)*x + (n^3 - 3*n^2 + 3*n - 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 theta_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidi…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • Affine, reversal and Möbius substitutions (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
        • 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 basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
                • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
                • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            no recorded property covers this requirement
        • ↑ Ma–Wang interval theorem (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Ma–Wang interval theorem (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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.