A038333 → x ↦ x+1 → basis A013609

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Submission text ?

Triangle begins:
  0:              676,               728,               196;
  1:            17576,             28392,             15288,              2744;
  2:           456976,            984256,            794976,            285376,              38416;
  3:         11881376,          31988320,          34448960,          18549440,            4994080,            537824;
  4:        308915776,         998035584,        1343509440,         964570880,          389538240,          83900544,           7529536;
  5:       8031810176,       30273746048,       48903743616,       43887975040,        23631986560,        7634949504,        1370375552,         105413504;
  6:     208827064576,      899562739712,     1695329778688,     1825739761664,      1228863301120,      529356498944,      142519057408,       21926008832,       1475789056;
  7:    5429503678976,    26312210136576,    56672452601856,    71203850704896,     57510802492416,    30967355188224,    11116486477824,     2565343033344,     345334639104,     20661046784;
  8:  141167095653376,   760130515056640,  1841854709560320,  2644714454753280,   2492134774671360,  1610302469787648,   722571621058560,   222329729556480,   44893503083520,   5371872163840,   289254654976;
  9: 3670344486987776, 21739732730619904, 58530049659361280, 94548541757429760, 101821506508001280, 76757751059877888, 41331096724549632, 15896575663288320, 4279847293962240, 768177719429120, 82726831323136, 4049565169664;

The row polynomials satisfy P(n,x) = (26+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 728*x + 196*x^2.

P[0, x_] := 676 + 728*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 14*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 676 + 728*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 14*x)*v[i - 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) 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 ?

• A013613 → x ↦ x+1 → x ↦ x/(x+1)
• A038335 → basis A013609 → x ↦ x/(x+1)
• A038284 → basis A013609 → x ↦ x/(x+1)
• A038321 → basis A013610 → basis A007318

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.