A159854 → basis A013611 → basis A007318

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Submission text ?

Triangle begins:
  0:        25,         40,         16;
  1:       150,        340,        256,          64;
  2:       900,       2640,       2896,        1408,         256;
  3:      5400,      19440,      27936,       20032,        7168,        1024;
  4:     32400,     138240,     245376,      231936,      123136,       34816,        4096;
  5:    194400,     959040,    2025216,     2373120,     1666560,      701440,      163840,      16384;
  6:   1166400,    6531840,   15987456,    22339584,    19491840,    10874880,     3788800,     753664,      65536;
  7:   6998400,   43856640,  122052096,   197987328,   206309376,   143216640,    66232320,   19677184,    3407872,    262144;
  8:  41990400,  291133440,  907739136,  1676132352,  2029805568,  1684537344,   970260480,  382992384,   99155968,  15204352,  1048576;
  9: 251942400, 1914762240, 6610968576, 13687750656, 18883362816, 18226446336, 12559712256, 6178996224, 2126905344, 487849984, 67108864, 4194304;

The row polynomials satisfy P(n,x) = (6+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 40*x + 16*x^2.

P[0, x_] := 25 + 40*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 25 + 40*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 4*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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (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_A013611 of Riordan array (1-x,x/(1-x)).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • 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
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of Riordan array (1-x,x/(1-x)).?
    • ↑ Geometric basis map (elementary) (above)

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.