basis-map coefficients of A112227 → basis A013611 → basis A013609

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Submission text ?

Triangle begins:
   1:       25,        80,         64;
   2:      125,       600,        960,         512;
   3:      625,      4000,       9600,       10240,         4096;
   4:     3125,     25000,      80000,      128000,       102400,        32768;
   5:    15625,    150000,     600000,     1280000,      1536000,       983040,       262144;
   6:    78125,    875000,    4200000,    11200000,     17920000,     17203200,      9175040,      2097152;
   7:   390625,   5000000,   28000000,    89600000,    179200000,    229376000,    183500800,     83886080,     16777216;
   8:  1953125,  28125000,  180000000,   672000000,   1612800000,   2580480000,   2752512000,   1887436800,    754974720,    134217728;
   9:  9765625, 156250000, 1125000000,  4800000000,  13440000000,  25804800000,  34406400000,  31457280000,  18874368000,   6710886400,  1073741824;
  10: 48828125, 859375000, 6875000000, 33000000000, 105600000000, 236544000000, 378470400000, 432537600000, 346030080000, 184549376000, 59055800320, 8589934592;

The row polynomials satisfy P(n,x) = (5+8*x)*P(n-1,x) for n >= 2, with P(1,x) = 25 + 80*x + 64*x^2.

P[1, x_] := 25 + 80*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 25 + 80*x + 64*x^2; for(n = 2, N, my(i = n); v[i] = (5 + 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, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • 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 A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.?
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of A scaled Hermite triangle.?
            • ↑ 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 basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.?
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of A scaled Hermite triangle.?
                • ↑ 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 A scaled Hermite triangle.?
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of A scaled Hermite triangle.?
                    • ↑ 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 A scaled Hermite triangle.?
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                    • Geometric basis map (elementary) polytool ? step ›

                      • from basis-map coefficients of A scaled Hermite triangle.?
                        • ↑ Parent-row provers of polytool (above)
                        • ↑ Iterated real-rootedness preservers (above)
                        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038271 → basis A013611 → basis A007318
• A136572 → basis A013611 → basis A013609
• A038220 → basis A013611 → D
• A130123 → basis A013611 → basis A013609
• A038336 → x ↦ x+1 → basis A013609
• A038319 → basis A013609 → D
• A038220 → basis A013611
• A038319 → basis A013609
• A013611 → D → basis A013609
• A120910 → basis A013609 → basis A013611
• A013622 → basis A007318
• A038319 → basis A013609 → reverse
• A038259 → basis A007318 → basis A013609
• basis-map coefficients of A026729 → basis A013611 → basis A013609
• A013622 → basis A007318 → D
• A038334 → basis A013611
• A038329 → basis A013610 → x ↦ 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.