A038284 → basis A013611 → basis A007318

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Submission text ?

Triangle begins:
  0:               1444,                1824,                 576;
  1:              54872,              103968,               65664,               13824;
  2:            2085136,             5267712,             4990464,             2101248,               331776;
  3:           79235168,           250216320,           316062720,           199618560,             63037440,              7962624;
  4:         3010936384,         11409864192,         18015575040,         15171010560,           7186268160,           1815478272,           191102976;
  5:       114415582592,        505837312512,        958428592128,       1008872202240,         637182443520,         241458610176,         50833391616,          4586471424;
  6:      4347792138496,      21967791857664,      48560382001152,      61339429896192,       48425865707520,       24467805831168,       7726675525632,       1394287312896,       110075314176;
  7:    165216101262848,     939123101915136,    2372521520627712,    3496347504082944,     3312329214394368,     2091997398564864,     880841009922048,     238423130505216,     37645757448192,      2641807540224;
  8:   6278211847988224,   39651864303083520,  112694772229816320,  189801721650216960,   209780850244976640,   158991802290929664,   83679895942594560,   30200263197327360,   7152693915156480,   1003886865285120,    63403380965376;
  9: 238572050223552512, 1657447927868891136, 5234046088007024640, 9917139956223836160, 12526913628914319360, 11076428892934766592, 6995639300800905216, 3155927504120709120, 996608685511802880, 209812354844590080, 26502613243527168, 1521681143169024;

The row polynomials satisfy P(n,x) = (38+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 1444 + 1824*x + 576*x^2.

P[0, x_] := 1444 + 1824*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(38 + 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] = 1444 + 1824*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (38 + 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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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 basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038305 → basis A013609 → basis A013609
• A038333 → x ↦ x/(x+1)
• A147716 → basis A013611 → 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.