A038305 → x ↦ x+1 → θ+1

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Submission text ?

Triangle begins:
  0:           169,           156,             27;
  1:          2197,          3042,           1053,            108;
  2:         28561,         52728,          27378,           5616,           405;
  3:        371293,        856830,         593190,         182520,         26325,          1458;
  4:       4826809,      13366548,       11567205,        4745520,       1026675,        113724,         5103;
  5:      62748517,     202725978,      210523131,      107960580,      31142475,       5174442,       464373,        17496;
  6:     815730721,    3011928816,     3649067604,     2245580064,     809704350,     179380656,     24147396,      1819584,       59049;
  7:   10604499373,   44049458934,    60991558524,    43788811248,   18947081790,    5246884188,    941748444,    106445664,     6908733,     196830;
  8:  137858491849,  636269962380,   991112826015,   813220780320,  410520105450,  136418988888,  30606824430,   4612645440,   449067645,   25587900,   649539;
  9: 1792160394037, 9098660462034, 15747681568905, 14536321448220, 8386339297050, 3251319235164, 875355178698, 164902074480, 21405557745, 1829534850, 92884077, 2125764;

The row polynomials satisfy (13+(3*n+6)*x)*P(n,x) = (169+(39*n+156)*x+(9*n+27)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 156*x + 27*x^2.

P[0, x_] := 169 + 156*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((169 + (39*n + 156)*x + (9*n + 27)*x^2)*P[n - 1, x])/(13 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 156*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((169 + (39*n + 156)*x + (9*n + 27)*x^2)*v[i - 1])/(13 + (3*n + 6)*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_A007318 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038327 → basis A013610 → θ+1

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.