A026729 → basis A013609 → θ+1

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Triangle begins:
   1:  1;
   2:  1,    4;
   3:  2,   12,    12;
   4:  3,   28,    60,    32;
   5:  5,   60,   192,   224,     80;
   6:  8,  120,   516,   928,    720,    192;
   7: 13,  232,  1248,  3040,   3680,   2112,    448;
   8: 21,  436,  2820,  8672,  14320,  12864,   5824,   1024;
   9: 34,  804,  6072, 22560,  47280,  58176,  41216,  15360,  2304;
  10: 55, 1460, 12612, 54944, 139680, 218880, 212800, 123904, 39168, 5120;

The row polynomials satisfy (20*n-20+(20*n^2+12*n-32)*x+(8*n^3+8*n^2-16*n)*x^2)*P(n,x) = -(40*n-100+(60*n^2-116*n-160)*x+(16*n^3-8*n^2-200*n-48)*x^2-(32*n^2+64*n)*x^3)*P(n-1,x) + (25+(50*n+90)*x+(20*n^2+160*n+92)*x^2+(48*n^2+136*n+24)*x^3+(16*n^2+32*n)*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[(-(40*n - 100 + (60*n^2 - 116*n - 160)*x + (16*n^3 - 8*n^2 - 200*n - 48)*x^2 - (32*n^2 + 64*n)*x^3)*P[n - 1, x] + (25 + (50*n + 90)*x + (20*n^2 + 160*n + 92)*x^2 + (48*n^2 + 136*n + 24)*x^3 + (16*n^2 + 32*n)*x^4)*D[P[n - 1, x], x])/(20*n - 20 + (20*n^2 + 12*n - 32)*x + (8*n^3 + 8*n^2 - 16*n)*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] = (-(40*n - 100 + (60*n^2 - 116*n - 160)*x + (16*n^3 - 8*n^2 - 200*n - 48)*x^2 - (32*n^2 + 64*n)*x^3)*v[i - 1] + (25 + (50*n + 90)*x + (20*n^2 + 160*n + 92)*x^2 + (48*n^2 + 136*n + 24)*x^3 + (16*n^2 + 32*n)*x^4)*deriv(v[i - 1], x))/(20*n - 20 + (20*n^2 + 12*n - 32)*x + (8*n^3 + 8*n^2 - 16*n)*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 basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward ant…?
    • Geometric basis map (elementary) 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)
    • Geometric basis map (elementary) 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)
    • Geometric basis map (elementary) 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.