A011973 → basis A013611 → D+1

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Triangle begins:
  0:   1;
  1:   1;
  2:   6,    4;
  3:  11,    8;
  4:  25,   52,   16;
  5:  48,  136,   48;
  6:  93,  368,  336,   64;
  7: 173,  856, 1120,  256;
  8: 318, 1916, 3504, 1920,  256;
  9: 575, 4072, 9456, 7680, 1280;

The row polynomials satisfy (64*n^3-112*n^2-28*n+(256*n^2-496*n)*x+256*n*x^2)*P(n,x) = (192*n^3-400*n^2-20*n+152+(512*n^3-640*n^2-1200*n+1088)*x+(2048*n^2-3200*n+1408)*x^2+(2048*n-2048)*x^3)*P(n-1,x) - (80*n^2+20*n-15+(576*n^2+464*n-88)*x+(1024*n^2+2560*n+272)*x^2+(4096*n+2560)*x^3+4096*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[((192*n^3 - 400*n^2 - 20*n + 152 + (512*n^3 - 640*n^2 - 1200*n + 1088)*x + (2048*n^2 - 3200*n + 1408)*x^2 + (2048*n - 2048)*x^3)*P[n - 1, x] - (80*n^2 + 20*n - 15 + (576*n^2 + 464*n - 88)*x + (1024*n^2 + 2560*n + 272)*x^2 + (4096*n + 2560)*x^3 + 4096*x^4)*D[P[n - 1, x], x])/(64*n^3 - 112*n^2 - 28*n + (256*n^2 - 496*n)*x + 256*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] = ((192*n^3 - 400*n^2 - 20*n + 152 + (512*n^3 - 640*n^2 - 1200*n + 1088)*x + (2048*n^2 - 3200*n + 1408)*x^2 + (2048*n - 2048)*x^3)*v[i - 1] - (80*n^2 + 20*n - 15 + (576*n^2 + 464*n - 88)*x + (1024*n^2 + 2560*n + 272)*x^2 + (4096*n + 2560)*x^3 + 4096*x^4)*deriv(v[i - 1], x))/(64*n^3 - 112*n^2 - 28*n + (256*n^2 - 496*n)*x + 256*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_A013611 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 A013611 → D+1
• A052553 → basis A013611 → 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.