A038331 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:       25,        170,         289;
  1:      125,       1275,        4335,         4913;
  2:      625,       8500,       43350,        98260,         83521;
  3:     3125,      53125,      361250,      1228250,       2088025,        1419857;
  4:    15625,     318750,     2709375,     12282500,      31320375,       42595710,       24137569;
  5:    78125,    1859375,    18965625,    107471875,     365404375,      745424925,      844814915,      410338673;
  6:   390625,   10625000,   126437500,    859775000,    3654043750,     9938999000,    16896298300,    16413546920,      6975757441;
  7:  1953125,   59765625,   812812500,   6448312500,   32886393750,   111813738750,   253444474500,   369304805700,    313909084845,    118587876497;
  8:  9765625,  332031250,  5080078125,  46059375000,  274053281250,  1118137387500,  3168055931250,  6155080095000,   7847727121125,   5929393824850,   2015993900449;
  9: 48828125, 1826171875, 31044921875, 316658203125, 2153275781250, 10249592718750, 34848615243750, 84632351306250, 143874997220625, 163058330183375, 110879664524695, 34271896307633;

The row polynomials satisfy P(n,x) = (5+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 170*x + 289*x^2.

P[0, x_] := 25 + 170*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 17*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 25 + 170*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 17*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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • 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)*5^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038331 → x ↦ x+1 → D
• A038331 → x ↦ x+1
• A013620 → basis A013611 → x ↦ x/(x+1)
• A038271 → basis A007318 → 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.