A038305 → x ↦ x+1 → basis A013610

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Submission text ?

Triangle begins:
  0:            256,             288,              81;
  1:           4096,            6912,            3888,             729;
  2:          65536,          147456,          124416,           46656,            6561;
  3:        1048576,         2949120,         3317760,         1866240,          524880,           59049;
  4:       16777216,        56623104,        79626240,        59719680,        25194240,         5668704,          531441;
  5:      268435456,      1056964608,      1783627776,      1672151040,       940584960,       317447424,        59521392,         4782969;
  6:     4294967296,     19327352832,     38050725888,     42807066624,     30098718720,     13544423424,      3809369088,       612220032,       43046721;
  7:    68719476736,    347892350976,    782757789696,   1027369598976,    866843099136,    487599243264,    182849716224,     44079842304,     6198727824,     387420489;
  8:  1099511627776,   6184752906240,  15655155793920,  23482733690880,  23115815976960,  15603175784448,   7313988648960,   2350924922880,   495898225920,   61987278240,   3486784401;
  9: 17592186044416, 108851651149824, 306145268858880, 516620141199360, 581197658849280, 457693156343808, 257452400443392, 103440696606720, 29092695920640, 5454880485120, 613674054576, 31381059609;

The row polynomials satisfy P(n,x) = (16+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 288*x + 81*x^2.

P[0, x_] := 256 + 288*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 9*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 288*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 9*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)*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 ?

• A038284 → basis A013610 → basis A007318
• A013611 → basis A007318 → basis A013611
• A038321 → basis A013609 → x ↦ x/(x+1)
• A038226 → basis A013609 → basis A013610
• A038335 → basis A013610 → basis A007318
• A038327 → basis A013610 → basis A013610

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.