A038331 → basis A013609 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:              729,               540,               100;
  1:            19683,             21870,              8100,              1000;
  2:           531441,            787320,            437400,            108000,             10000;
  3:         14348907,          26572050,          19683000,           7290000,           1350000,            100000;
  4:        387420489,         860934420,         797161500,         393660000,         109350000,          16200000,          1000000;
  5:      10460353203,       27119434230,       30132704700,       18600435000,        6889050000,        1530900000,        189000000,         10000000;
  6:     282429536481,      836828256240,     1084777369200,      803538792000,      372008700000,      110224800000,      20412000000,       2160000000,       100000000;
  7:    7625597484987,    25418658283290,    37657271530800,    32543321076000,    18079622820000,     6696156600000,    1653372000000,     262440000000,     24300000000,     1000000000;
  8:  205891132094649,   762559748498700,  1270932914164500,  1255242384360000,   813583026900000,   361592456400000,  111602610000000,   23619600000000,   3280500000000,   270000000000,   10000000000;
  9: 5559060566555523, 22648024530411390, 41940786167428500, 46600873519365000, 34519165569900000, 17898826591800000, 6629195034000000, 1753755300000000, 324769500000000, 40095000000000, 2970000000000, 100000000000;

The row polynomials satisfy P(n,x) = (27+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 540*x + 100*x^2.

P[0, x_] := 729 + 540*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 729 + 540*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 10*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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • 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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • ↑ Geometric basis map (elementary) (above)

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.