basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1

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Triangle begins:
  0:       1;
  1:      -4,       -3;
  2:      16,       24,        9;
  3:     -64,     -144,     -108,      -27;
  4:     256,      768,      864,      432,        81;
  5:   -1024,    -3840,    -5760,    -4320,     -1620,     -243;
  6:    4096,    18432,    34560,    34560,     19440,     5832,      729;
  7:  -16384,   -86016,  -193536,  -241920,   -181440,   -81648,   -20412,    -2187;
  8:   65536,   393216,  1032192,  1548288,   1451520,   870912,   326592,    69984,    6561;
  9: -262144, -1769472, -5308416, -9289728, -10450944, -7838208, -3919104, -1259712, -236196, -19683;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4 + 3*x)*P[n - 1, 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 + 3*x)*v[i - 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) 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)
• Affine, reversal and Möbius substitutions (elementary) 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) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A027465 → D → basis A013610
• A013610 → basis A007318
• A178822 → x ↦ x+1 → basis A013611
• A038335 → reverse
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• A085880 → x ↦ x+1 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• A178820 → basis A007318 → basis A013611
• A130123 → basis A013610 → x ↦ x+1
• A120910 → basis A013611 → D
• A038284 → reverse
• A038330 → basis A013610 → D
• A038330 → basis A013610
• A038284 → D
• A013610 → basis A013611 → x ↦ x+1
• A178821 → basis A007318 → basis A013611
• A120910 → basis A013611
• A038330 → x ↦ x/(x+1)
• A038334 → basis A013609 → basis A013610
• A094305 → x ↦ x+1 → basis A013611
• A356546 → x ↦ x+1 → basis A013611
• A130123 → basis A013609 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A196347 → x ↦ x+1 → basis A013611
• A038329 → basis A013610 → basis A013609
• A120910 → D → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• A196347 → basis A013611 → basis A013609
• A038221 → basis A013611 → basis A013609
• A038279 → basis A013610 → x ↦ x/(x+1)
• A085881 → basis A013611 → basis A013609
• A038208 → basis A013611 → basis A013609
• A038330 → basis A013610 → reverse
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• A028326 → basis A007318 → basis A013611
• A038329 → basis A013611 → basis A013610
• A013609 → D → basis A013609
• A178822 → basis A013611 → basis A013609
• A156992 → basis A013611 → basis A013609
• A356546 → basis A013611 → basis A013609
• A178820 → basis A013611 → basis A013609
• A028326 → basis A013611 → basis A013609
• A013609 → basis A013609
• A120909 → basis A013610 → basis A013609
• A038335 → D
• A120910 → basis A013611 → reverse
• A027465 → x ↦ x/(x+1)
• A013609 → basis A013609 → D
• A136572 → basis A013610 → basis A007318
• A085880 → basis A013611 → basis A013609
• A135065 → x ↦ x+1 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• basis-map coefficients of A026729 → basis A013610 → basis A007318
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
• A013610 → basis A007318 → D
• A003506 → x ↦ x+1 → basis A013611
• A038243 → basis A013609 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013610 → basis A007318
• A178821 → basis A013611 → 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.