A038305 → basis A013609 → θ+1

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Submission text ?

Triangle begins:
  0:           169,            312,            108;
  1:          2197,           6084,           4212,             864;
  2:         28561,         105456,         109512,           44928,            6480;
  3:        371293,        1713660,        2372760,         1460160,          421200,           46656;
  4:       4826809,       26733096,       46268820,        37964160,        16426800,         3639168,         326592;
  5:      62748517,      405451956,      842092524,       863684640,       498279600,       165582144,       29719872,        2239488;
  6:     815730721,     6023857632,    14596270416,     17964640512,     12955269600,      5740180992,     1545433344,      232906752,      15116544;
  7:   10604499373,    88098917868,   243966234096,    350310489984,    303153308640,    167900294016,    60271900416,    13625044992,    1768635648,    100776960;
  8:  137858491849,  1272539924760,  3964451304060,   6505766242560,   6568321687200,   4365407644416,  1958836763520,   590418616320,  114961317120,  13101004800,   665127936;
  9: 1792160394037, 18197320924068, 62990726275620, 116290571585760, 134181428752800, 104042215525248, 56022731436672, 21107465533440, 5479822782720, 936721843200, 95113294848, 4353564672;

The row polynomials satisfy (13+(6*n+12)*x)*P(n,x) = (169+(78*n+312)*x+(36*n+108)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 312*x + 108*x^2.

P[0, x_] := 169 + 312*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((169 + (78*n + 312)*x + (36*n + 108)*x^2)*P[n - 1, x])/(13 + (6*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 312*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((169 + (78*n + 312)*x + (36*n + 108)*x^2)*v[i - 1])/(13 + (6*n + 12)*x)); 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 (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check

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.