A038321 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           154,             98;
  1:          2541,           3234,           1029;
  2:         37268,          71148,          45276,           9604;
  3:        512435,        1304380,        1245090,         528220,          84035;
  4:       6764142,       21522270,       27391980,       17431260,        5546310,         705894;
  5:      86806489,      331442958,      527295615,      447402340,      213532935,       54353838,        5764801;
  6:    1091281576,     4861163384,     9280402824,     9842851480,     6263632760,     2391568872,      507302488,       46118408;
  7:   13504609503,    68750739288,   153126646596,   194888459304,   155024910810,    78921772776,    25111473156,     4565722392,     363182463;
  8:  165056338370,   945322665210,  2406275875080,  3572955087240,  3410548037820,  2170348751340,   920754015720,   251114731560,   39950070930,   2824752490;
  9: 1997181694277, 12709338054490, 36394922610585, 61761080793720, 68779385429370, 52522439782428, 27852808975530, 10128294172920, 2416979291265, 341795051290, 21750594173;

The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(7*n+14)*x)*P(n-1,x) for n >= 1, with P(0,x) = 154 + 98*x.

P[0, x_] := 154 + 98*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (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] = 154 + 98*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (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)*11^(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 ?

• A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• A038301 → basis A013609 → basis A013610
• A038321 → reverse
• A038328 → basis A013611 → x ↦ x/(x+1)
• A038277 → reverse
• A038277 → D

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.