basis-map coefficients of A030528, shifted → basis A013609 → basis A013611

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Triangle begins:
  0:      1;
  1:     -3,      -8;
  2:      9,      48,       64;
  3:    -27,    -216,     -576,      -512;
  4:     81,     864,     3456,      6144,       4096;
  5:   -243,   -3240,   -17280,    -46080,     -61440,     -32768;
  6:    729,   11664,    77760,    276480,     552960,     589824,     262144;
  7:  -2187,  -40824,  -326592,  -1451520,   -3870720,   -6193152,   -5505024,   -2097152;
  8:   6561,  139968,  1306368,   6967296,   23224320,   49545216,   66060288,   50331648,   16777216;
  9: -19683, -472392, -5038848, -31352832, -125411328, -334430208, -594542592, -679477248, -452984832, -134217728;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(3 + 8*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] = -(3 + 8*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 (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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 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 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 ?

• A130123 → basis A013609 → basis A013611
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A028326 → basis A013611 → basis A013611
• A013612 → basis A013609 → x ↦ x/(x+1)
• A013609 → basis A013610 → basis A013611
• A038329 → basis A013610 → basis A007318
• A027466 → basis A013610
• A136572 → basis A013609 → basis A013611
• A038208 → basis A013611 → basis A013611
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• basis-map coefficients of A026729 → basis A013609 → basis A013611
• A038328 → x ↦ x+1 → basis A013610
• A120909 → basis A013610 → basis A013611
• A120909 → basis A013611
• A013622 → x ↦ x/(x+1)
• A013622 → x ↦ x/(x+1) → D
• A013609 → D → basis A013611
• A178821 → basis A013611 → basis A013611
• A038281 → D
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• A038226 → D
• A178820 → basis A013611 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → basis A013611
• A038327 → basis A013610 → basis A013609
• A038271 → basis A013611 → x ↦ x/(x+1)
• A027466 → basis A013610 → D

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.