A038328 → basis A013609 → reverse

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Submission text ?

Triangle begins:
  0:      16,       112,        196;
  1:      64,       672,       2352,        2744;
  2:     256,      3584,      18816,       43904,        38416;
  3:    1024,     17920,     125440,      439040,       768320,        537824;
  4:    4096,     86016,     752640,     3512320,      9219840,      12907776,       7529536;
  5:   16384,    401408,    4214784,    24586240,     86051840,     180708864,     210827008,     105413504;
  6:   65536,   1835008,   22478848,   157351936,    688414720,    1927561216,    3373232128,    3373232128,     1475789056;
  7:  262144,   8257536,  115605504,   944111616,   4956585984,   17348050944,   40478785536,   60718178304,    53128406016,    20661046784;
  8: 1048576,  36700160,  578027520,  5394923520,  33043906560,  138784407552,  404787855360,  809575710720,  1062568120320,   826441871360,   289254654976;
  9: 4194304, 161480704, 2825912320, 29672079360, 207704555520, 1017752322048, 3562133127168, 8905332817920, 15584332431360, 18181721169920, 12727204818944, 4049565169664;

The row polynomials satisfy P(n,x) = (4+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + 112*x + 196*x^2.

P[0, x_] := 16 + 112*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 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] = 16 + 112*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 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_A013609 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

Related ?

Same fingerprint ?

• A013612 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A013611 → x ↦ x/(x+1)
• A038291 → basis A013611 → basis A007318
• A038255 → basis A013609 → reverse
• A038329 → basis A013609 → basis A007318
• A038255 → basis A013609 → D
• A038334 → x ↦ x+1 → x ↦ x+1
• A038328 → basis A013609
• A038255 → basis A013609
• A038328 → basis A013609 → D

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.