A131436 → D+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 3,  3;
  2: 0, 14,  7;
  3: 0,  0, 45,  15;
  4: 0,  0,  0, 124,  31;
  5: 0,  0,  0,   0, 315,  63;
  6: 0,  0,  0,   0,   0, 762,  127;
  7: 0,  0,  0,   0,   0,   0, 1785,  255;
  8: 0,  0,  0,   0,   0,   0,    0, 4088,  511;
  9: 0,  0,  0,   0,   0,   0,    0,    0, 9207, 1023;

The row polynomials satisfy (n^2-3*n+2+(2*n-3)*x+x^2)*P(n,x) = -((2*n^2-2*n)*x^2+(4*n-2)*x^3+2*x^4)*P(n-2,x) + ((3*n^2-6*n)*x+(6*n-6)*x^2+3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((2*n^2 - 2*n)*x^2 + (4*n - 2)*x^3 + 2*x^4)*P[n - 2, x] + ((3*n^2 - 6*n)*x + (6*n - 6)*x^2 + 3*x^3)*P[n - 1, x])/(n^2 - 3*n + 2 + (2*n - 3)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-((2*n^2 - 2*n)*x^2 + (4*n - 2)*x^3 + 2*x^4)*v[i - 2] + ((3*n^2 - 6*n)*x + (6*n - 6)*x^2 + 3*x^3)*v[i - 1])/(n^2 - 3*n + 2 + (2*n - 3)*x + 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 Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
    • 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 Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A130123 → D+1
• A131431 → θ+1 → D+1
• A130123 → D → D+1
• image under θ+1 → θ+1 → D+1 → reverse
• A131431 → (θ+1)² → D+1
• image under θ+1 → θ+1 → D+1
• A136572 → θ+1 → D+1
• A127773 → D+1
• basis-map coefficients of A026729, shifted → θ+1 → D+1
• Hadamard-factor parent of A127773 → D+1
• A127773 → D → D+1
• A127773 → θ+1 → D+1
• A136572 → D → D+1
• A127648 → D → D+1
• basis-map coefficients of A030528, shifted → D+1
• basis-map coefficients of A026729, shifted → D+1
• basis-map coefficients of A026729, shifted → (θ+1)² → D+1
• A131431 → D+1 → reverse
• basis-map coefficients of A026729 → D+1
• A132440 → θ+1 → D+1
• basis-map coefficients of A021012 → D+1
• basis-map coefficients of A021012 → θ+1 → D+1
• A131431 → D+1
• A127648 → (θ+1)² → D+1
• basis-map coefficients of A030528, shifted → D → D+1
• A131431 → D+1 → D
• A132440 → D+1
• image under θ+1 → (θ+1)² → D+1
• basis-map coefficients of A026729 → (θ+1)² → D+1
• A136572 → (θ+1)² → D+1
• basis-map coefficients of A026729, shifted → D → D+1
• image under θ+1 → θ+1 → D+1 → D
• image under θ+1 → D+1
• basis-map coefficients of A030528, shifted → θ+1 → D+1
• A130123 → θ+1 → D+1
• basis-map coefficients of A030528, shifted → (θ+1)² → D+1
• A130123 → (θ+1)² → D+1
• A136572 → D+1
• A023531 → D+1
• basis-map coefficients of A026729 → 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.