A038305 → basis A013610 → basis A007318

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Submission text ?

Triangle begins:
  0:             484,              396,               81;
  1:           10648,            13068,             5346,              729;
  2:          234256,           383328,           235224,            64152,             6561;
  3:         5153632,         10541520,          8624880,          3528360,           721710,            59049;
  4:       113379904,        278296128,        284621040,        155247840,         47632860,          7794468,           531441;
  5:      2494357888,       7142933952,       8766328032,       5977041840,       2445153480,        600174036,         81841914,         4782969;
  6:     54875873536,     179593767936,     257145622272,     210391872768,     107586753120,      35210210112,       7202088432,       841802544,       43046721;
  7:   1207269217792,    4444945756416,    7273547601408,    6942931801344,    4260435423552,    1742905400544,     475337836512,     83338451856,     8523250758,      387420489;
  8:  26559922791424,  108654229601280,  200022559038720,  218206428042240,  156215965530240,   76687837623936,   26143581008160,   6111486469440,   937557583380,    85232507580,   3486784401;
  9: 584318301411328, 2629432356350976, 5378384365263360, 6600744448277760, 5400609094045440, 3093076117498752, 1265349320794944, 369744931401120, 75629645059320, 10313133417180, 843801825042, 31381059609;

The row polynomials satisfy P(n,x) = (22+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 396*x + 81*x^2.

P[0, x_] := 484 + 396*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 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] = 484 + 396*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 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 (−∞, -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)*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 basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038277 → basis A013611

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.