A038321 → basis A013611 → basis A007318

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Submission text ?

Triangle begins:
  0:                2116,                 2576,                  784;
  1:               97336,               177744,               108192,                21952;
  2:             4477456,             10901632,              9953664,              4039168,               614656;
  3:           205962976,            626843840,            763114240,            464504320,            141370880,             17210368;
  4:          9474296896,          34601779968,          52654882560,          42734397440,          19509181440,           4750061568,            481890304;
  5:        435817657216,        1856962191616,        3390974436864,        3440118993920,        2093985474560,         764759912448,         155168677888,          13492928512;
  6:      20047612231936,       97623155216384,      207979765460992,      253192757952512,      192646663659520,       93810549260288,       28551036731392,        4965397692416,        377801998336;
  7:     922190162669056,     5051998282447872,    12300517557264384,    17470300298723328,    15951143751008256,     9709391848439808,     3940043068932096,     1027837322330112,     156410027311104,      10578455953408;
  8:   42420747482776576,   258213245547335680,   707279759542702080,  1148048305344675840,  1222921020910632960,   893264050056462336,   453104952927191040,   157601722757283840,   35974306281553920,    4866089738567680,    296196766695424;
  9: 1951354384207722496, 13065590224695185408, 39764839814289694720, 72614055313050746880, 88399719511540039680, 75331934888094990336, 45854221236231733248, 19936617928796405760, 6067666326155427840, 1231120703857623040, 149875563947884544, 8293509467471872;

The row polynomials satisfy P(n,x) = (46+28*x)*P(n-1,x) for n >= 1, with P(0,x) = 2116 + 2576*x + 784*x^2.

P[0, x_] := 2116 + 2576*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[(46 + 28*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2116 + 2576*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = (46 + 28*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 Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • 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
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • ↑ 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.