A038328 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:    4,     56,     196;
  1:    8,    168,    1176,      2744;
  2:   16,    448,    4704,     21952,      38416;
  3:   32,   1120,   15680,    109760,     384160,      537824;
  4:   64,   2688,   47040,    439040,    2304960,     6453888,      7529536;
  5:  128,   6272,  131712,   1536640,   10756480,    45177216,    105413504,    105413504;
  6:  256,  14336,  351232,   4917248,   43025920,   240945152,    843308032,   1686616064,    1475789056;
  7:  512,  32256,  903168,  14751744,  154893312,  1084253184,   5059848192,  15179544576,   26564203008,   20661046784;
  8: 1024,  71680, 2257920,  42147840,  516311040,  4337012736,  25299240960, 101196963840,  265642030080,  413220935680,  289254654976;
  9: 2048, 157696, 5519360, 115906560, 1622691840, 15902380032, 111316660224, 556583301120, 1948041553920, 4545430292480, 6363602409472, 4049565169664;

The row polynomials satisfy P(n,x) = (2+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 56*x + 196*x^2.

P[0, x_] := 4 + 56*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 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] = 4 + 56*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 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 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)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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)*2^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/(x+1)
• A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).
• A027466 → D
• A038327 → x ↦ x+1 → basis A013609
• A027466 → reverse
• A038328 → x ↦ x+1 → D
• A038328 → x ↦ x+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.