A038329 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:            270,             162;
  1:           6075,            7290,            2187;
  2:         121500,          218700,          131220,            26244;
  3:        2278125,         5467500,         4920750,          1968300,           295245;
  4:       41006250,       123018750,       147622500,         88573500,         26572050,          3188646;
  5:      717609375,      2583393750,      3875090625,       3100072500,       1395032625,        334807830,        33480783;
  6:    12301875000,     51667875000,     93002175000,      93002175000,      55801305000,      20088469800,      4017693960,       344373768;
  7:   207594140625,    996451875000,   2092548937500,    2511058725000,    1883294043750,     903981141000,    271194342300,     46490458680,     3486784401;
  8:  3459902343750,  18683472656250,  44840334375000,   62776468125000,   56498821312500,   33899292787500,  13559717115000,   3486784401000,   523017660150,   34867844010;
  9: 57088388671875, 342530332031250, 924831896484375, 1479731034375000, 1553717586093750, 1118676661987500, 559338330993750, 191773142055000, 43148956962375, 5753194261650, 345191655699;

The row polynomials satisfy (n+1)*P(n,x) = (15*n+30+(9*n+18)*x)*P(n-1,x) for n >= 1, with P(0,x) = 270 + 162*x.

P[0, x_] := 270 + 162*x;
P[n_, x_] := P[n, x] = Expand[Together[((15*n + 30 + (9*n + 18)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 270 + 162*x; for(n = 1, N, my(i = n + 1); v[i] = ((15*n + 30 + (9*n + 18)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 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

Related ?

Same fingerprint ?

• A178820 → basis A013610 → x ↦ x+1
• A013620 → basis A007318 → D
• A038221 → basis A013610 → basis A007318
• A178822 → basis A013610 → basis A007318
• A120910 → basis A013609 → x ↦ x/(x+1)
• A013622 → D
• A038271 → basis A013611
• A038334 → x ↦ x/(x+1)
• A013612 → basis A013609 → reverse
• A038329 → basis A013610 → reverse
• A038255 → basis A013610 → basis A013609
• A028326 → basis A013610 → x ↦ x+1
• A038243 → basis A013611 → x ↦ x/(x+1)
• A038284 → basis A007318 → basis A013609
• A038271 → basis A013610 → basis A013610
• A027467 → basis A013611 → basis A013610
• A013620 → basis A007318
• A013612 → basis A013609 → D
• A178821 → basis A013610 → x ↦ x+1
• A013612 → basis A013609
• A038271 → basis A013611 → D
• A038338 → basis A013610 → x ↦ x+1
• A038231 → D → basis A013610
• A038335 → x ↦ x+1 → basis A013609
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
• A085880 → basis A013610 → x ↦ x+1
• A013622 → basis A013609 → basis A013610
• A156992 → basis A013610 → x ↦ x+1
• A038329 → basis A013610
• A038319 → basis A013611 → basis A013610
• A196347 → basis A013610 → basis A007318

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.