A038333 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:             266,               98;
  1:            7581,             5586,             1029;
  2:          192052,           212268,            78204,             9604;
  3:         4561235,          6721820,          3714690,           912380,            84035;
  4:       103996158,        191571870,        141158220,         52005660,          9579990,          705894;
  5:      2305248169,       5095811742,       4693510815,       2305584260,        637069335,        93883902,         5764801;
  6:     50056817384,     129093897464,     142682728776,      87612201880,      32278179640,      7135176552,       876249752,       46118408;
  7:   1069964471583,    3153579495192,    4066457770116,    2996337304296,    1379892179610,    406705063464,     74919353796,     7886247768,     363182463;
  8:  22588138844530,   74897513010810,  110375282331720,   94884014636040,   52435902825180,  19318490514540,   4744892407080,   749193537960,   69004667970,   2824752490;
  9: 472092101850677, 1739286691028810, 2883554250916185, 2832965579847480, 1826517281743770, 807512903507772, 247920628269930, 52193816477880, 7210987802865, 590373270410, 21750594173;

The row polynomials satisfy (n+1)*P(n,x) = (19*n+38+(7*n+14)*x)*P(n-1,x) for n >= 1, with P(0,x) = 266 + 98*x.

P[0, x_] := 266 + 98*x;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 38 + (7*n + 14)*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] = 266 + 98*x; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 38 + (7*n + 14)*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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038271 → x ↦ x/(x+1) → x ↦ x/(x+1)
• A038284 → basis A013611 → x ↦ x/(x+1)
• A038333 → x ↦ x+1
• A038333 → x ↦ x+1 → reverse

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.