A038334 → basis A013610 → basis A007318

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:                1936,                2112,                  576;
  1:               85184,              139392,                76032,                13824;
  2:             3748096,             8177664,              6690816,              2433024,               331776;
  3:           164916224,           449771520,            490659840,            267632640,             72990720,              7962624;
  4:          7256313856,         23747936256,          32383549440,          23551672320,           9634775040,           2102132736,            191102976;
  5:        319277809664,       1219060727808,        1994826645504,        1813478768640,         989170237440,         323728441344,          58859716608,          4586471424;
  6:      14048223625216,      61301339455488,      117029829869568,      127668905312256,       87046980894720,       37984137117696,       10359310123008,       1614437941248,        110075314176;
  7:     618121839509504,    3034416303046656,     6620544661192704,     8426147750608896,     6894120886861824,     3760429574651904,     1367428936237056,     319658712367104,      43589824413696,      2641807540224;
  8:   27197360938418176,  148349241482280960,   364129956365598720,   529643572895416320,   505568865036533760,   330917802569367552,   150417182986076160,   46883277813841920,    9589761371013120,   1162395317698560,    63403380965376;
  9: 1196683881290399744, 7180103287742398464, 19582099875661086720, 32043436160172687360, 34956475811097477120, 26694036073928982528, 14560383313052172288, 5672876615474872320, 1547148167856783360, 281299666883051520, 30687236387241984, 1521681143169024;

The row polynomials satisfy P(n,x) = (44+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 1936 + 2112*x + 576*x^2.

P[0, x_] := 1936 + 2112*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(44 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1936 + 2112*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (44 + 24*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 (−∞, -1]
  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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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 basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038320 → reverse
• A013612 → x ↦ x+1 → x ↦ x/(x+1)
• A038271 → basis A013609 → x ↦ x/(x+1)
• A038277 → basis A013610
• A038336 → x ↦ x/(x+1) → D
• A038314 → x ↦ x+1 → D
• A027466 → basis A013610 → basis A013609
• A038320 → D
• A038336 → x ↦ x/(x+1)
• A038259 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• A038314 → 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.