A038333 → D

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Submission text ?

Triangle begins:
  0:           168,             98;
  1:          3024,           3528,           1029;
  2:         48384,          84672,          49392,            9604;
  3:        725760,        1693440,        1481760,          576240,           84035;
  4:      10450944,       30481920,       35562240,        20744640,         6050520,         705894;
  5:     146313216,      512096256,      746807040,       580849920,       254121840,       59295096,        5764801;
  6:    2006581248,     8193540096,    14338695168,     13940398080,      8131898880,     2846164608,      553420896,       46118408;
  7:   27088846848,   126414618624,   258096513024,    301112598528,    219561269760,   102461925888,    29884728384,     4980788064,     363182463;
  8:  361184624640,  1896219279360,  4424511651840,   6022251970560,   5269470474240,  3073857776640,  1195389135360,   298847283840,   43581895560,   2824752490;
  9: 4767637045248, 27811216097280, 73004442255360, 113562465730560, 115928350433280, 81149845303296, 39447841466880, 13149280488960, 2876405106960, 372867328680, 21750594173;

The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(7*n+14)*x)*P(n-1,x) for n >= 1, with P(0,x) = 168 + 98*x.

P[0, x_] := 168 + 98*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (7*n + 14)*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] = 168 + 98*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (7*n + 14)*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 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 ?

• A038271 → x ↦ x/(x+1) → reverse
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• A038333 → reverse
• A038284 → basis A013611
• A038284 → basis A013611 → reverse
• A038335 → basis A013611
• A038271 → x ↦ x/(x+1) → D
• A038284 → basis A013611 → D
• A038271 → x ↦ x/(x+1)
• A120910 → 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.