basis-map coefficients of A030528, shifted → basis A013610 → D

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Triangle begins:
  0:   0;
  1:  -3;
  2:   6,   18;
  3:  -9,  -54,   -81;
  4:  12,  108,   324,    324;
  5: -15, -180,  -810,  -1620,   -1215;
  6:  18,  270,  1620,   4860,    7290,    4374;
  7: -21, -378, -2835, -11340,  -25515,  -30618,  -15309;
  8:  24,  504,  4536,  22680,   68040,  122472,  122472,   52488;
  9: -27, -648, -6804, -40824, -153090, -367416, -551124, -472392, -177147;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = -3; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 3*n*x)*v[i - 1])/(n - 1)); 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_A013610 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 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)
    • 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) 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 (above)
                        • ↑ Iterated real-rootedness preservers (above)
                        • ↑ Affine, reversal and Möbius substitutions (elementary) (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) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
• A038221 → basis A013611 → x ↦ x/(x+1)
• A130123 → θ+1 → basis A013610
• A131436 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A156992 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → θ+1 → basis A013610
• image under θ+1 → θ+1 → D → basis A013610
• A120910 → D → x ↦ x+1
• A196347 → x ↦ x+1 → basis A007318
• A028326 → basis A007318 → basis A007318
• A136572 → basis A013610 → D
• A134309 → basis A013610
• basis-map coefficients of A112227 → basis A013610
• A120909 → basis A013610 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → (θ+1)² → basis A013610
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A038279 → basis A013610 → reverse
• A131431 → θ+1 → basis A013610
• A028326 → basis A013611 → x ↦ x/(x+1)
• A038330 → D
• A131431 → D → basis A013610
• A130123 → basis A013610
• A085881 → basis A013611 → x ↦ x/(x+1)
• A120910 → basis A007318
• A130123 → (θ+1)² → basis A013610
• A085880 → x ↦ x+1 → x ↦ x+1
• A356546 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A013610 → D
• A013609 → basis A013610 → x ↦ x/(x+1)
• A178821 → basis A007318 → x ↦ x+1
• A038329 → x ↦ x+1 → basis A013609
• A038330 → basis A013609 → x ↦ x+1
• basis-map coefficients of A026729 → D → basis A013610
• A038243 → basis A013609
• basis-map coefficients of A030528, shifted → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
• A129186 → basis A013610
• basis-map coefficients of A021012 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → D → basis A013610
• basis-map coefficients of A112227 → θ+1 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → reverse
• A136572 → basis A013610 → basis A013611
• A013610 → D → D
• basis-map coefficients of A112227 → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610
• image under θ+1 → (θ+1)² → basis A013610
• A013610 → D
• A013610 → basis A013611 → D
• A127648 → D → basis A013610
• A038243 → basis A013609 → D
• A038330 → reverse
• A038279 → basis A013610 → D
• A178821 → basis A013611 → x ↦ x/(x+1)
• A356546 → basis A013611 → x ↦ x/(x+1)
• A196347 → basis A013611 → x ↦ x/(x+1)
• image under θ+1 → basis A013610
• basis-map coefficients of A026729, shifted → D → basis A013610
• A127648 → basis A013610
• A136572 → (θ+1)² → basis A013610
• A136572 → D → basis A013610
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
• basis-map coefficients of A026729 → basis A013610 → D
• basis-map coefficients of A026729 → basis A013610
• A120909 → x ↦ x/(x+1)
• A027465 → D → D
• basis-map coefficients of A026729 → basis A013610 → reverse
• A027465: Cube of lower triangular normalized binomial matrix.
• A038279 → basis A013610
• A130123 → basis A013610 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
• A136572 → θ+1 → basis A013610
• A136572 → basis A013610
• A178822 → x ↦ x+1 → x ↦ x+1
• A178820 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → (θ+1)² → basis A013610
• A132440 → θ+1 → basis A013610
• A085881 → basis A007318 → basis A007318
• A120910 → basis A007318 → reverse
• A178820 → basis A007318 → basis A007318
• A131431 → basis A013610
• A038325 → basis A007318 → basis A007318
• A094305 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A021012 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → D
• A085880 → basis A013611 → x ↦ x/(x+1)
• A135065 → x ↦ x+1 → basis A007318
• A027465 → D
• A136521 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
• basis-map coefficients of A026729 → basis A013610 → basis A013611
• A013610 → basis A013611 → reverse
• basis-map coefficients of A026729, shifted → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
• A013610 → D → basis A013611
• A013610 → basis A013611
• A130123 → basis A013610 → D
• A130123 → D → basis A013610
• A178822 → basis A013611 → x ↦ x/(x+1)
• A038338 → x ↦ x+1 → basis A007318
• A147716 → basis A013610 → basis A013609
• A156992 → basis A007318 → x ↦ x+1

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.