A038305 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:            312,             288;
  1:           6084,           11232,            5184;
  2:         105456,          292032,          269568,            82944;
  3:        1713660,         6327360,         8760960,          5391360,          1244160;
  4:       26733096,       123383520,       227784960,        210263040,         97044480,         17915904;
  5:      405451956,      2245580064,      5182107840,       6377978880,       4415523840,       1630347264,        250822656;
  6:     6023857632,     38923387776,    107787843072,     165827450880,     153071493120,      84778057728,      26085556224,       3439853568;
  7:    88098917868,    650576624256,   2101862939904,    3880362350592,    4477341173760,    3306344251392,    1526005039104,     402462867456,     46438023168;
  8:  1272539924760,  10571870144160,  39034597455360,   84074517596160,  116410870517760,  107456188170240,   66126885027840,   26160086384640,   6036943011840,   619173642240;
  9: 18197320924068, 167975270068320, 697743429514560, 1717522288035840, 2774459080673280, 3073246981668864, 2364036139745280, 1246964117667840, 431641425346560, 88541830840320, 8173092077568;

The row polynomials satisfy (n+1)*P(n,x) = (13*n+26+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 312 + 288*x.

P[0, x_] := 312 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + 26 + (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] = 312 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((13*n + 26 + (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_A013611 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

Related ?

Same fingerprint ?

• A038305 → basis A013611
• A038327 → x ↦ x/(x+1)
• A038305 → basis A013611 → reverse
• A038335 → basis A013609 → basis A013609
• A038284 → basis A013609 → basis A013609
• A038279 → basis A013611 → basis A013610

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.