A038284 → basis A013609 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:            336,             288;
  1:           7056,           12096,             5184;
  2:         131712,          338688,           290304,            82944;
  3:        2304960,         7902720,         10160640,          5806080,          1244160;
  4:       38723328,       165957120,        284497920,        243855360,        104509440,         17915904;
  5:      632481024,      3252759552,       6970199040,       7965941760,       5120962560,       1755758592,        250822656;
  6:    10119696384,     60718178304,     156132458496,     223046369280,     191182602240,      98322481152,      28092137472,       3439853568;
  7:   159385218048,   1092927209472,    3278781628416,    5620768505856,    6022251970560,    4129544208384,    1769804660736,     433421549568,     46438023168;
  8:  2479325614080,  19126226165760,   65575632568320,  131151265136640,  168623055175680,  144534047293440,   82590884167680,   30339508469760,   6501323243520,   619173642240;
  9: 38181614456832, 327270981058560, 1262330926940160, 2885327833006080, 4327991749509120, 4451648656637952, 3179749040455680, 1557428101447680, 500601889751040, 95352740904960, 8173092077568;

The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 336 + 288*x.

P[0, x_] := 336 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (12*n + 24)*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] = 336 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (12*n + 24)*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A038321 → basis A013610 → D
• A038329 → basis A013609 → basis A013610
• A038335 → basis A013609 → reverse
• A013613 → x ↦ x+1 → reverse
• A038328 → x ↦ x/(x+1)
• A038335 → basis A013609
• A013613 → x ↦ x+1
• A038284 → basis A013609 → reverse
• A038335 → basis A013609 → D
• A038255 → x ↦ x/(x+1)
• A038321 → basis A013610
• A038291 → basis A013611 → basis A013610
• A038330 → basis A013610 → basis A013609
• A038284 → basis A013609
• A013613 → x ↦ x+1 → D

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.