A134346 → θ+1

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Triangle begins:
  0:    1;
  1:    3,     6;
  2:    7,    28,     21;
  3:   15,    90,    135,     60;
  4:   31,   248,    558,    496,    155;
  5:   63,   630,   1890,   2520,   1575,    378;
  6:  127,  1524,   5715,  10160,   9525,   4572,    889;
  7:  255,  3570,  16065,  35700,  44625,  32130,  12495,   2040;
  8:  511,  8176,  42924, 114464, 178850, 171696, 100156,  32704,  4599;
  9: 1023, 18414, 110484, 343728, 644490, 773388, 601524, 294624, 82863, 10230;

The row polynomials satisfy (1+(2*n-1)*x+(n^2-n)*x^2)*P(n,x) = -(2+(4*n+6)*x+(2*n^2+10*n+6)*x^2+(4*n^2+8*n+2)*x^3+(2*n^2+2*n)*x^4)*P(n-2,x) + (3+(6*n+3)*x+(3*n^2+6*n-3)*x^2+(3*n^2-3)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 + (4*n + 6)*x + (2*n^2 + 10*n + 6)*x^2 + (4*n^2 + 8*n + 2)*x^3 + (2*n^2 + 2*n)*x^4)*P[n - 2, x] + (3 + (6*n + 3)*x + (3*n^2 + 6*n - 3)*x^2 + (3*n^2 - 3)*x^3)*P[n - 1, x])/(1 + (2*n - 1)*x + (n^2 - n)*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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (4*n + 6)*x + (2*n^2 + 10*n + 6)*x^2 + (4*n^2 + 8*n + 2)*x^3 + (2*n^2 + 2*n)*x^4)*v[i - 2] + (3 + (6*n + 3)*x + (3*n^2 + 6*n - 3)*x^2 + (3*n^2 - 3)*x^3)*v[i - 1])/(1 + (2*n - 1)*x + (n^2 - 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
    • 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 [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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A085880 → θ+1
• A135065 → θ+1
• A038330 → basis A013611 → D+1
• A085880 → D → θ+1
• A131431 → x ↦ x+1 → D+1
• A120910 → basis A013610 → θ+1
• A085881 → basis A013609 → D+1
• A038325 → θ+1
• A038221 → basis A013609 → D+1
• A093375 → D → D
• A038208 → D → θ+1
• A156992 → basis A013609 → θ+1
• A038221 → D+1
• A216973 → θ+1
• A027465 → basis A013611 → D+1
• A038338 → basis A013609 → θ+1
• A028326 → basis A013609 → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A178821 → basis A013609 → θ+1
• A178820 → θ+1
• A135838 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A003506 → D → θ+1
• A038338 → basis A013609 → D+1
• A196347 → θ+1
• A038325 → basis A013609 → D+1
• A134346 → D+1
• basis-map coefficients of A113214, shifted → D → θ+1
• basis-map coefficients of A113214, shifted → θ+1
• A038338 → θ+1
• A085881 → basis A013609 → θ+1
• A038221 → θ+1
• A038325 → D+1
• basis-map coefficients of A113214, shifted → D+1
• A085880 → D+1
• A038330 → basis A013611 → θ+1
• A085880 → basis A013609 → θ+1
• A038208 → D → D+1
• A038338 → D+1
• A085880 → basis A013609 → D+1
• A027465 → basis A013611 → θ+1
• A094305 → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• basis-map coefficients of A026729 → basis A007318 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A124860 → θ+1
• A131431 → x ↦ x+1 → θ+1
• A344565 → θ+1
• A344565 → D+1
• A085881 → θ+1
• A135065 → D+1
• A178821 → D+1
• A376827 → θ+1
• A124860 → D+1
• A093375 → D
• A131431 → D+1 → basis A007318
• A196347 → basis A013609 → θ+1
• A356546 → basis A013609 → θ+1
• basis-map coefficients of A127647 → D+1
• A028326 → θ+1
• A356546 → basis A013609 → D+1
• basis-map coefficients of A127647 → θ+1
• A156992 → D+1
• A356546 → D+1
• A028326 → D → D+1
• A178822 → basis A013609 → θ+1
• A028326 → D → θ+1
• A120910 → basis A013610 → D+1
• A178822 → θ+1
• A356546 → θ+1
• A085880 → D → D+1
• A156992 → θ+1
• A178820 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A028326 → D+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A038208 → θ+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A178822 → D+1
• A196347 → D+1
• A028326 → basis A013609 → D+1
• A038325 → basis A013609 → θ+1
• A178822 → basis A013609 → D+1
• basis-map coefficients of A113214, shifted → D → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A156992 → basis A013609 → D+1
• A178820 → basis A013609 → D+1
• A178821 → basis A013609 → D+1
• A196347 → basis A013609 → D+1
• A085881 → D+1
• A376827 → D+1
• A094305 → D+1
• A038208 → D+1
• A038221 → basis A013609 → θ+1
• A135838 → θ+1
• A178820 → basis A013609 → θ+1
• A178821 → θ+1

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.