A038329 → x ↦ x+1 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:            324,             108,              9;
  1:           5832,            2916,            486,             27;
  2:         104976,           69984,          17496,           1944,             81;
  3:        1889568,         1574640,         524880,          87480,           7290,           243;
  4:       34012224,        34012224,       14171760,        3149280,         393660,         26244,          729;
  5:      612220032,       714256704,      357128352,       99202320,       16533720,       1653372,        91854,        2187;
  6:    11019960576,     14693280768,     8571080448,     2857026816,      595213920,      79361856,      6613488,      314928,       6561;
  7:   198359290368,    297538935552,   198359290368,    77139724032,    19284931008,    3214155168,    357128352,    25509168,    1062882,     19683;
  8:  3570467226624,   5950778711040,  4463084033280,  1983592903680,   578547930240,  115709586048,  16070775840,  1530550080,   95659380,   3542940,    59049;
  9: 64268410079232, 117825418478592, 98187848732160, 49093924366080, 16364641455360, 3818416339584, 636402723264, 75762228960, 6313519080, 350751060, 11691702, 177147;

The row polynomials satisfy P(n,x) = (18+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 108*x + 9*x^2.

P[0, x_] := 324 + 108*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 108*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 3*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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
• A013613 → D
• A038255 → D
• A013612 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• A038328 → D
• A038328 → reverse
• A038291 → basis A013609 → basis A007318
• A147716 → basis A013610 → x ↦ x+1
• A013613: Triangle of coefficients in expansion of (1+6x)^n.

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.