A013618 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:            792,            2178;
  1:          14256,           78408,           107811;
  2:         228096,         1881792,          5174928,           4743684;
  3:        3421440,        37635840,        155247840,         284621040,          195676965;
  4:       49268736,       677445120,       3725948160,       10246357440,        14088741480,         7748807814;
  5:      689762304,     11381078016,      78244911360,      286898008320,       591727142160,       650899856376,        298329100839;
  6:     9459597312,    182097248256,    1502302298112,     6885552199680,     18935268549120,     31243193106048,      28639593680544,      11251268945928;
  7:   127704563712,   2809500401664,   27041441366016,   148727927513088,    511252250826240,   1124754951817728,    1546538058749376,    1215137046160224,     417703359617577;
  8:  1702727516160,  42142506024960,  463567566274560,  2974558550261760,  12270054019829760,  33742648554531840,   61861522349975040,   72908222769613440,   50124403154109240,   15315789852644490;
  9: 22476003213312, 618090088366080, 7648864843530240, 56091675519221760, 269941188436254720, 890805921839640576, 2041430237549176320, 3207961801862991360, 3308210608171209840, 2021684260549072680, 555963171650994987;

The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(33*n+66)*x)*P(n-1,x) for n >= 1, with P(0,x) = 792 + 2178*x.

P[0, x_] := 792 + 2178*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (33*n + 66)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 792 + 2178*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (33*n + 66)*x)*v[i - 1])/(n + 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 (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+11x)^n.?
        • 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

Related ?

Same fingerprint ?

• A038328 → basis A013611 → basis A007318
• A038303 → basis A013611 → reverse
• A013618 → basis A013610
• A013618 → basis A013610 → reverse
• A038303 → basis A013611
• A038303 → 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. [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.