basis-map coefficients of A112227 → basis A013610 → basis A013610

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1:      16,        72,         81;
   2:      64,       432,        972,        729;
   3:     256,      2304,       7776,      11664,        6561;
   4:    1024,     11520,      51840,     116640,      131220,        59049;
   5:    4096,     55296,     311040,     933120,     1574640,      1417176,       531441;
   6:   16384,    258048,    1741824,    6531840,    14696640,     19840464,     14880348,      4782969;
   7:   65536,   1179648,    9289728,   41803776,   117573120,    211631616,    238085568,    153055008,     43046721;
   8:  262144,   5308416,   47775744,  250822656,   846526464,   1904684544,   2857026816,   2754990144,   1549681956,    387420489;
   9: 1048576,  23592960,  238878720, 1433272320,  5643509760,  15237476352,  28570268160,  36733201920,  30993639120,  15496819560,   3486784401;
  10: 4194304, 103809024, 1167851520, 7882997760, 35473489920, 111741493248, 251418359808, 404065221120, 454573373760, 340930030320, 153418513644, 31381059609;

The row polynomials satisfy P(n,x) = (4+9*x)*P(n-1,x) for n >= 2, with P(1,x) = 16 + 72*x + 81*x^2.

P[1, x_] := 16 + 72*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 9*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 16 + 72*x + 81*x^2; for(n = 2, N, my(i = n); v[i] = (4 + 9*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_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 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_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 ?

• basis-map coefficients of A030528, shifted → basis A013610 → basis A013610
• A013610 → basis A013610 → D
• A038328 → basis A013609 → basis A013609
• A013622 → basis A013609 → x ↦ x/(x+1)
• A038279 → basis A013611
• A038329 → basis A013611 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013610 → basis A013610
• A136572 → basis A013610 → basis A013610
• A038271 → basis A013610 → x ↦ x/(x+1)
• A038334 → basis A013609 → basis A007318
• A013610 → D → basis A013610
• A130123 → basis A013610 → basis A013610
• A038319 → basis A013611 → x ↦ x/(x+1)
• A013610 → basis A013611 → basis A013610
• A013610 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013610
• A038279 → basis A013611 → 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.