A038329 → x ↦ x+1 → θ+1

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Submission text ?

Triangle begins:
  0:           225,            180,             27;
  1:          3375,           4050,           1215,            108;
  2:         50625,          81000,          36450,           6480,            405;
  3:        759375,        1518750,         911250,         243000,          30375,          1458;
  4:      11390625,       27337500,       20503125,        7290000,        1366875,        131220,          5103;
  5:     170859375,      478406250,      430565625,      191362500,       47840625,       6889050,        535815,        17496;
  6:    2562890625,     8201250000,     8611312500,     4592700000,     1435218750,     275562000,      32148900,      2099520,       59049;
  7:   38443359375,   138396093750,   166075312500,   103335750000,    38750906250,    9300217500,    1446700500,    141717600,     7971615,     196830;
  8:  576650390625,  2306601562500,  3113912109375,  2214337500000,   968772656250,  279006525000,   54251268750,   7085880000,   597871125,   29524500,    649539;
  9: 8649755859375, 38058925781250, 57088388671875, 45670710937500, 22835355468750, 7672679437500, 1790291868750, 292292550000, 32882911875, 2435771250, 107173935, 2125764;

The row polynomials satisfy (5+(n+2)*x)*P(n,x) = (75+(15*n+60)*x+(3*n+9)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 180*x + 27*x^2.

P[0, x_] := 225 + 180*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((75 + (15*n + 60)*x + (3*n + 9)*x^2)*P[n - 1, x])/(5 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 180*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((75 + (15*n + 60)*x + (3*n + 9)*x^2)*v[i - 1])/(5 + (n + 2)*x)); 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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)

Related ?

Same fingerprint ?

• A038243 → θ+1
• A147716 → basis A013610 → θ+1
• A038291 → basis A013609 → θ+1

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.