A038329 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           216,            90,             9;
  1:          3024,          1944,           405,            27;
  2:         41472,         36288,         11664,          1620,            81;
  3:        559872,        622080,        272160,         58320,          6075,          243;
  4:       7464960,      10077696,       5598720,       1632960,        262440,        21870,          729;
  5:      98537472,     156764160,     105815808,      39191040,       8573040,      1102248,        76545,        2187;
  6:    1289945088,    2364899328,    1881169920,     846526464,     235146240,     41150592,      4408992,      262440,       6561;
  7:   16769286144,   34828517376,   31926140928,   16930529280,    5714053632,   1269789696,    185177664,    17006112,     885735,     19683;
  8:  216710774784,  503078584320,  522427760640,  319261409280,  126978969600,  34284321792,   6348948480,   793618560,   63772920,   2952450,   59049;
  9: 2786281390080, 7151455567872, 8300796641280, 5746705367040, 2633906626560, 838061199360, 188563769856, 29930757120, 3273676560, 233834040, 9743085, 177147;

The row polynomials satisfy (n+5+x)*P(n,x) = (12*n+72+(3*n+30)*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 216 + 90*x + 9*x^2.

P[0, x_] := 216 + 90*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 72 + (3*n + 30)*x + 3*x^2)*P[n - 1, x])/(n + 5 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 216 + 90*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 72 + (3*n + 30)*x + 3*x^2)*v[i - 1])/(n + 5 + 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 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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(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 ?

• A027466 → basis A013609 → D+1
• A027467 → basis A013611 → D+1
• A038330 → basis A007318 → D+1
• A038231 → D → D+1
• A038231 → D+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.