A038305 → basis A013609 → basis A007318

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Submission text ?

Triangle begins:
  0:             361,             228,              36;
  1:            6859,            6498,            2052,             216;
  2:          130321,          164616,           77976,           16416,            1296;
  3:         2476099,         3909630,         2469240,          779760,          123120,            7776;
  4:        47045881,        89139564,        70373340,        29630880,         7017840,          886464,          46656;
  5:       893871739,      1975927002,      1871930844,       985226760,       311124240,        58949856,        6205248,         279936;
  6:     16983563041,     42905843472,     47422248048,     29950893504,     11822721120,      2986792704,      471598848,       42550272,       1679616;
  7:    322687697779,    917112404214,   1158457773744,    853600464864,    404337062304,    127685388096,    26881134336,     3638048256,     287214336,     10077696;
  8:   6131066257801,  19361261866740,  27513372126420,  23169155474880,  12804006972960,   4852044747648,  1276853880960,   230409722880,   27285361920,   1914762240,    60466176;
  9: 116490258898219, 404650373014866, 638921641602420, 605294186781240, 382291065335520, 169012892043072, 53372492224128, 12038908020480, 1900880213760, 200092654080, 12637430784, 362797056;

The row polynomials satisfy P(n,x) = (19+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 228*x + 36*x^2.

P[0, x_] := 361 + 228*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 228*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 6*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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (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_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
• 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) (above)

Related ?

Same fingerprint ?

• A027467 → basis A013610 → 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.