A038321 → basis A007318 → reverse

Matches ?

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Submission text ?

Triangle begins:
  0:         49,         252,          324;
  1:        343,        2646,         6804,          5832;
  2:       2401,       24696,        95256,        163296,         104976;
  3:      16807,      216090,      1111320,       2857680,        3674160,         1889568;
  4:     117649,     1815156,     11668860,      40007520,       77157360,        79361856,        34012224;
  5:     823543,    14823774,    114354828,     490092120,     1260236880,      1944365472,      1666598976,       612220032;
  6:    5764801,   118590192,   1067311728,    5489031744,    17643316320,     36294822144,     46664771328,     34284321792,     11019960576;
  7:   40353607,   933897762,   9605805552,   57634833312,   222305785632,    571643448768,    979960197888,   1079956136448,    694257516288,    198359290368;
  8:  282475249,  7263649260,  84050798580,  576348333120,  2593567499040,   8003008282752,  17149303463040,  25198976517120,  24299013070080,  13885150325760,   3570467226624;
  9: 1977326743, 55930099302, 719101276740, 5547352706280, 28529242489440, 102705272961984, 264099273330816, 485080297954560, 623674668798720, 534578287541760, 274925976450048, 64268410079232;

The row polynomials satisfy P(n,x) = (7+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 49 + 252*x + 324*x^2.

P[0, x_] := 49 + 252*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(7 + 18*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 49 + 252*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + 18*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 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)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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 Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038277 → x ↦ x/(x+1) → D
• A038277 → x ↦ x/(x+1) → reverse
• A038277 → x ↦ x/(x+1)
• A013620 → basis A013610 → basis A013611
• A038321 → basis A007318
• A038321 → basis A007318 → D

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.