A038305 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:      9,      78,       169;
  1:     27,     351,      1521,       2197;
  2:     81,    1404,      9126,      26364,       28561;
  3:    243,    5265,     45630,     197730,      428415,       371293;
  4:    729,   18954,    205335,    1186380,     3855735,      6683274,      4826809;
  5:   2187,   66339,    862407,    6228495,    26990145,     70174377,    101362989,      62748517;
  6:   6561,  227448,   3449628,   29896776,   161940870,    561395016,   1216355868,    1505964408,     815730721;
  7:  19683,  767637,  13305708,  134535492,   874480698,   3789416358,  10947202812,   20330519508,   22024729467,   10604499373;
  8:  59049, 2558790,  49896405,  576580680,  4372403490,  22736498148,  82104021090,  203305195080,  330370942005,  318134981190,  137858491849;
  9: 177147, 8444007, 182953485, 2378395305, 20612759310, 125050739814, 541886539194, 1677267859410, 3634080362055, 5249227189635, 4549330231017, 1792160394037;

The row polynomials satisfy P(n,x) = (3+13*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 78*x + 169*x^2.

P[0, x_] := 9 + 78*x + 169*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 9 + 78*x + 169*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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_A007318 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 Triangle whose (i,j)-th entry is binomial(i,j)*10^(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)

Related ?

Same fingerprint ?

• A038327 → basis A013610 → D
• A038305 → x ↦ x+1 → D
• A038305 → x ↦ x+1
• A013612 → basis A013611 → x ↦ x/(x+1)
• A038327 → basis A013610
• A038327 → basis A013610 → 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.