A038328 → basis A013609 → θ+1

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Submission text ?

Triangle begins:
  0:           196,            224,             48;
  1:          2744,           4704,           2016,            256;
  2:         38416,          87808,          56448,          14336,           1280;
  3:        537824,        1536640,        1317120,         501760,          89600,           6144;
  4:       7529536,       25815552,       27659520,       14049280,        3763200,         516096,         28672;
  5:     105413504,      421654016,      542126592,      344207360,      122931200,       25288704,       2809856,        131072;
  6:    1475789056,     6746464256,    10119696384,     7710244864,     3442073600,      944111616,     157351936,      14680064,       589824;
  7:   20661046784,   106256812032,   182154534912,   161915142144,    86740254720,    29739515904,    6608781312,     924844032,     74317824,     2621440;
  8:  289254654976,  1652883742720,  3187704360960,  3238302842880,  2023939276800,   832706445312,  231307345920,   43159388160,   5202247680,   367001600,   11534336;
  9: 4049565169664, 25454409637888, 54545163509760, 62337329725440, 44526664089600, 21372798763008, 7124266254336, 1661636444160, 267048714240, 28259123200, 1776287744, 50331648;

The row polynomials satisfy (7+(2*n+4)*x)*P(n,x) = (98+(28*n+112)*x+(8*n+24)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 224*x + 48*x^2.

P[0, x_] := 196 + 224*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (28*n + 112)*x + (8*n + 24)*x^2)*P[n - 1, x])/(7 + (2*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 224*x + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (28*n + 112)*x + (8*n + 24)*x^2)*v[i - 1])/(7 + (2*n + 4)*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 basis_map_A013609 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 ?

• A038255 → basis A013609 → θ+1

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.