A011973 → D → D+1

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Triangle begins:
  0:  0;
  1:  0;
  2:  1;
  3:  2;
  4:  5,   2;
  5: 10,   6;
  6: 17,  18,   3;
  7: 26,  44,  12;
  8: 37,  90,  42,  4;
  9: 50, 162, 120, 20;

The row polynomials satisfy (n^3-8*n^2+22*n-20+(4*n^2-25*n+34)*x+(4*n-8)*x^2)*P(n,x) = (n^3-6*n^2+13*n-10+(2*n^3-14*n^2+33*n-37)*x+(8*n^2-46*n+74)*x^2+(8*n-24)*x^3)*P(n-1,x) - ((n^2-4*n+5)*x+(4*n^2-12*n+11)*x^2+(16*n-32)*x^3+16*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 - 6*n^2 + 13*n - 10 + (2*n^3 - 14*n^2 + 33*n - 37)*x + (8*n^2 - 46*n + 74)*x^2 + (8*n - 24)*x^3)*P[n - 1, x] - ((n^2 - 4*n + 5)*x + (4*n^2 - 12*n + 11)*x^2 + (16*n - 32)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - 8*n^2 + 22*n - 20 + (4*n^2 - 25*n + 34)*x + (4*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 - 6*n^2 + 13*n - 10 + (2*n^3 - 14*n^2 + 33*n - 37)*x + (8*n^2 - 46*n + 74)*x^2 + (8*n - 24)*x^3)*v[i - 1] - ((n^2 - 4*n + 5)*x + (4*n^2 - 12*n + 11)*x^2 + (16*n - 32)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - 8*n^2 + 22*n - 20 + (4*n^2 - 25*n + 34)*x + (4*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 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 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 ?

• A169803 → D → D+1
• A052553 → D → 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.