basis-map coefficients of A113214, shifted → x ↦ x+1 → θ+1

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Triangle begins:
  0:    1;
  1:   -2,    -2;
  2:    4,     8,      3;
  3:   -8,   -24,    -18,     -4;
  4:   16,    64,     72,     32,      5;
  5:  -32,  -160,   -240,   -160,    -50,     -6;
  6:   64,   384,    720,    640,    300,     72,     7;
  7: -128,  -896,  -2016,  -2240,  -1400,   -504,   -98,    -8;
  8:  256,  2048,   5376,   7168,   5600,   2688,   784,   128,    9;
  9: -512, -4608, -13824, -21504, -20160, -12096, -4704, -1152, -162, -10;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4 + (2*n + 4)*x + (n + 1)*x^2)*P[n - 1, x])/(2 + n*x)]];
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] = (-(4 + (2*n + 4)*x + (n + 1)*x^2)*v[i - 1])/(2 + n*x)); 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 shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
                • Parent-row provers of polytool polytool ? step › direct check
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A120910 → θ+1
• A156992 → basis A007318 → θ+1
• A196347 → x ↦ x+1 → θ+1
• A120910 → D → θ+1
• A356546 → x ↦ x+1 → θ+1
• A038338 → x ↦ x+1 → θ+1
• A038330 → basis A013609 → θ+1
• A132440 → D+1 → basis A013609
• A178822 → x ↦ x+1 → θ+1
• A085881 → basis A007318 → θ+1
• A178821 → basis A007318 → θ+1
• A038208 → basis A007318 → θ+1
• A027466 → basis A013611 → θ+1
• A003506 → x ↦ x+1 → θ+1
• A131431 → D+1 → basis A013609
• A028326 → basis A007318 → θ+1
• A085880 → x ↦ x+1 → θ+1
• A094305 → x ↦ x+1 → θ+1
• A135065 → x ↦ x+1 → θ+1
• A178820 → basis A007318 → θ+1
• A038325 → basis A007318 → θ+1
• A038207 → θ+1
• A038221 → x ↦ x+1 → θ+1
• image under θ+1 → θ+1 → D+1 → basis A013609

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.