A038305 → basis A013609 → reverse

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Submission text ?

Triangle begins:
  0:        36,        156,         169;
  1:       216,       1404,        3042,         2197;
  2:      1296,      11232,       36504,        52728,         28561;
  3:      7776,      84240,      365040,       790920,        856830,        371293;
  4:     46656,     606528,     3285360,      9491040,      15422940,      13366548,        4826809;
  5:    279936,    4245696,    27597024,     99655920,     215921160,     280697508,      202725978,       62748517;
  6:   1679616,   29113344,   220776192,    956696832,    2591053920,    4491160128,     4865423472,     3011928816,      815730721;
  7:  10077696,  196515072,  1703130624,   8610271488,   27983382336,   60630661728,    87577622496,    81322078032,    44049458934,    10604499373;
  8:  60466176, 1310100480, 12773479680,  73802327040,  279833823360,  727567940736,  1313664337440,  1626441560640,  1321483768020,   636269962380,  137858491849;
  9: 362797056, 8646663168, 93672184320, 608869198080, 2638433191680, 8003247348096, 17340369254208, 26836285750560, 29072642896440, 20996908758540, 9098660462034, 1792160394037;

The row polynomials satisfy P(n,x) = (6+13*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 156*x + 169*x^2.

P[0, x_] := 36 + 156*x + 169*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 13*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 156*x + 169*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 13*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_A013609 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

Related ?

Same fingerprint ?

• A038284 → basis A013609 → basis A007318
• A038321 → basis A013610 → x ↦ x/(x+1)
• A038335 → basis A013609 → x ↦ x+1
• A038305 → basis A013609 → D
• A038291 → basis A013610 → basis A013609
• A038305 → basis A013609

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.