A038305 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:            234,            162;
  1:           4563,           6318,            2187;
  2:          79092,         164268,          113724,           26244;
  3:        1285245,        3559140,         3696030,         1705860,          295245;
  4:       20049822,       69403230,        96096780,        66528540,        23029110,         3188646;
  5:      304088967,     1263138786,      2186201745,      2018032380,      1047824505,       290166786,        33480783;
  6:     4517893224,    21894405624,     45472996296,     52468841880,     36324582840,     15088672872,      3482001432,       344373768;
  7:    66074188401,   365949351144,    886723427772,   1227770899992,   1062494048070,    588458242008,    203697083772,     40291730856,     3486784401;
  8:   954404943570,  5946676956090,  16467720801480,  26601702833160,  27624845249820,  19124892865260,   8826873630120,   2618962505640,   453281972130,   34867844010;
  9: 13647990693051, 94486089413430, 294360509326455, 543434786448840, 658392145120710, 546971935946436, 315560732276790, 124837212768840, 32409661007295, 4986101693430, 345191655699;

The row polynomials satisfy (n+1)*P(n,x) = (13*n+26+(9*n+18)*x)*P(n-1,x) for n >= 1, with P(0,x) = 234 + 162*x.

P[0, x_] := 234 + 162*x;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + 26 + (9*n + 18)*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] = 234 + 162*x; for(n = 1, N, my(i = n + 1); v[i] = ((13*n + 26 + (9*n + 18)*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 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

Related ?

Same fingerprint ?

• A013610 → basis A013610 → x ↦ x+1
• A038279 → basis A013611 → x ↦ x/(x+1)
• A038305 → basis A013610 → reverse
• A038305 → basis A013610

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.