A038333 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:             532,              392;
  1:           15162,            22344,              8232;
  2:          384104,           849072,            625632,            153664;
  3:         9122470,         26887280,          29717520,          14598080,           2689120;
  4:       207992316,        766287480,        1129265760,         832090560,         306559680,          45177216;
  5:      4610496338,      20383246968,       37548086520,       36889348160,       20386218720,        6008569728,         737894528;
  6:    100113634768,     516375589856,     1141461830208,     1401795230080,     1032901748480,      456651299328,      112159968256,       11806312448;
  7:   2139928943166,   12614317980768,    32531662160928,    47941396868736,    44156549747520,    26029124061696,     9589677285888,     2018879428608,     185949421056;
  8:  45176277689060,  299590052043240,   883002258653760,  1518144234176640,  1677948890405760,  1236383392930560,   607346228106240,   191793545717760,   35330390000640,   2892546549760;
  9: 944184203701354, 6957146764115240, 23068434007329480, 45327449277559680, 58448553015800640, 51680825824497408, 31733840418551040, 13361617018337280, 3692025755066880, 604542228899840, 44545216866304;

The row polynomials satisfy (n+1)*P(n,x) = (19*n+38+(14*n+28)*x)*P(n-1,x) for n >= 1, with P(0,x) = 532 + 392*x.

P[0, x_] := 532 + 392*x;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 38 + (14*n + 28)*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] = 532 + 392*x; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 38 + (14*n + 28)*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)*12^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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 ?

• A038333 → basis A013609
• A038333 → basis A013609 → reverse

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.