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

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Submission text ?

Triangle begins:
   1:       25,        160,         256;
   2:      125,       1200,        3840,         4096;
   3:      625,       8000,       38400,        81920,         65536;
   4:     3125,      50000,      320000,      1024000,       1638400,       1048576;
   5:    15625,     300000,     2400000,     10240000,      24576000,      31457280,       16777216;
   6:    78125,    1750000,    16800000,     89600000,     286720000,     550502400,      587202560,      268435456;
   7:   390625,   10000000,   112000000,    716800000,    2867200000,    7340032000,    11744051200,    10737418240,     4294967296;
   8:  1953125,   56250000,   720000000,   5376000000,   25804800000,   82575360000,   176160768000,   241591910400,   193273528320,    68719476736;
   9:  9765625,  312500000,  4500000000,  38400000000,  215040000000,  825753600000,  2202009600000,  4026531840000,  4831838208000,  3435973836800,  1099511627776;
  10: 48828125, 1718750000, 27500000000, 264000000000, 1689600000000, 7569408000000, 24222105600000, 55364812800000, 88583700480000, 94489280512000, 60473139527680, 17592186044416;

The row polynomials satisfy P(n,x) = (5+16*x)*P(n-1,x) for n >= 2, with P(1,x) = 25 + 160*x + 256*x^2.

P[1, x_] := 25 + 160*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 25 + 160*x + 256*x^2; for(n = 2, N, my(i = n); v[i] = (5 + 16*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 ?

• A013611 → D → basis A013611
• basis-map coefficients of A026729 → basis A013611 → basis A013611
• A038301 → basis A013611 → x ↦ x/(x+1)
• A038319 → x ↦ x+1 → D
• A038336 → x ↦ x+1 → basis A007318
• A038319 → x ↦ x+1
• A013611 → basis A013611 → D
• A130123 → basis A013611 → basis A013611
• A013611 → basis A013611
• A038319 → x ↦ x+1 → reverse
• A038259 → 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.