A038333 → reverse

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Submission text ?

Triangle begins:
  0:         49,         168,          144;
  1:        343,        1764,         3024,          1728;
  2:       2401,       16464,        42336,         48384,         20736;
  3:      16807,      144060,       493920,        846720,        725760,         248832;
  4:     117649,     1210104,      5186160,      11854080,      15240960,       10450944,        2985984;
  5:     823543,     9882516,     50824368,     145212480,     248935680,      256048128,      146313216,       35831808;
  6:    5764801,    79060128,    474360768,    1626379776,    3485099520,     4779565056,     4096770048,     2006581248,      429981696;
  7:   40353607,   622598508,   4269246912,   17076987648,   43912253952,    75278149632,    86032171008,    63207309312,    27088846848,     5159780352;
  8:  282475249,  4842432840,  37355910480,  170769876480,  512309629440,  1053894094848,  1505562992640,  1474837217280,   948109639680,   361184624640,   61917364224;
  9: 1977326743, 37286732868, 319600567440, 1643660061120, 5635405923840, 13524974217216, 23185670086656, 28390616432640, 24334814085120, 13905608048640, 4767637045248, 743008370688;

The row polynomials satisfy P(n,x) = (7+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 49 + 168*x + 144*x^2.

P[0, x_] := 49 + 168*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(7 + 12*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 + 168*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + 12*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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(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

Related ?

Same fingerprint ?

• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• A038284 → basis A013611
• A120910 → basis A013611 → basis A013610
• A038271 → x ↦ x/(x+1)
• A038271 → x ↦ x/(x+1) → D
• A038271 → x ↦ x/(x+1) → reverse
• A038333 → D
• A038335 → basis A013611
• A038284 → basis A013611 → D
• A038284 → basis A013611 → reverse

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.