A038328 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:            224,             128;
  1:           4704,            5376,            1536;
  2:          87808,          150528,           86016,           16384;
  3:        1536640,         3512320,         3010560,         1146880,          163840;
  4:       25815552,        73758720,        84295680,        48168960,        13762560,         1572864;
  5:      421654016,      1445670912,      2065244160,      1573519360,       674365440,       154140672,        14680064;
  6:     6746464256,     26985857024,     46261469184,     44058542080,     25176309760,      8631877632,      1644167168,      134217728;
  7:   106256812032,    485745426432,    971490852864,   1110275260416,    793053757440,    362538860544,    103582531584,    16911433728,     1207959552;
  8:  1652883742720,   8500544962560,  19429817057280,  25906422743040,  22205505208320,  12688860119040,   4833851473920,  1183800360960,   169114337280,   10737418240;
  9: 25454409637888, 145453769359360, 374023978352640, 569941300346880, 569941300346880, 390816891666432, 186103281745920, 60768418529280, 13021803970560, 1653562408960, 94489280512;

The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(8*n+16)*x)*P(n-1,x) for n >= 1, with P(0,x) = 224 + 128*x.

P[0, x_] := 224 + 128*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (8*n + 16)*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] = 224 + 128*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (8*n + 16)*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)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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 ?

• A038329 → basis A013609 → basis A013609
• A013610 → basis A007318 → x ↦ x/(x+1)
• A038255 → basis A013611 → D
• A120910 → basis A013611 → x ↦ x+1
• A038255 → basis A013611
• A038334 → x ↦ x+1 → basis A013609
• A038328 → basis A013611
• A038284 → x ↦ x/(x+1)
• A038335 → x ↦ x/(x+1)
• A038330 → basis A013610 → x ↦ x/(x+1)
• A038255 → basis A013611 → reverse
• A038328 → basis A013611 → reverse
• A038291 → basis A013611 → basis A013609

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.