A038329 → θ+1

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Submission text ?

Triangle begins:
  0:          144,           144,            27;
  1:         1728,          2592,           972,           108;
  2:        20736,         41472,         23328,          5184,           405;
  3:       248832,        622080,        466560,        155520,         24300,          1458;
  4:      2985984,       8957952,       8398080,       3732480,        874800,        104976,         5103;
  5:     35831808,     125411328,     141087744,      78382080,      24494400,       4408992,       428652,        17496;
  6:    429981696,    1719926784,    2257403904,    1504935936,     587865600,     141087744,     20575296,      1679616,       59049;
  7:   5159780352,   23219011584,   34828517376,   27088846848,   12697896960,    3809369088,    740710656,     90699264,     6377292,     196830;
  8:  61917364224,  309586821120,  522427760640,  464380231680,  253957939200,   91424858112,  22221319680,   3627970560,   382637520,   23619600,   649539;
  9: 743008370688, 4086546038784, 7662273822720, 7662273822720, 4788921139200, 2011346878464, 586642839552, 119723028480, 16836050880, 1558893600, 85739148, 2125764;

The row polynomials satisfy (4+(n+2)*x)*P(n,x) = (48+(12*n+48)*x+(3*n+9)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 144*x + 27*x^2.

P[0, x_] := 144 + 144*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((48 + (12*n + 48)*x + (3*n + 9)*x^2)*P[n - 1, x])/(4 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 144*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48 + (12*n + 48)*x + (3*n + 9)*x^2)*v[i - 1])/(4 + (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 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

Related ?

Same fingerprint ?

• A027466 → basis A013609 → θ+1
• A038330 → basis A007318 → θ+1
• A038231 → D → θ+1
• A131431 → D+1 → basis A013611
• A132440 → D+1 → basis A013611
• A038231 → θ+1
• A027467 → basis A013611 → θ+1
• image under θ+1 → θ+1 → D+1 → basis A013611

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.