A109983 → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0: 1;
  1: 0, 2, 6;
  2: 0, 0, 3, 24, 30;
  3: 0, 0, 0,  4, 60, 180, 140;
  4: 0, 0, 0,  0,  5, 120, 630, 1120,  630;
  5: 0, 0, 0,  0,  0,   6, 210, 1680, 5040,  6300,  2772;
  6: 0, 0, 0,  0,  0,   0,   7,  336, 3780, 16800, 34650,  33264,  12012;
  7: 0, 0, 0,  0,  0,   0,   0,    8,  504,  7560, 46200, 138600, 216216,  168168,   51480;
  8: 0, 0, 0,  0,  0,   0,   0,    0,    9,   720, 13860, 110880, 450450, 1009008, 1261260,  823680,  218790;
  9: 0, 0, 0,  0,  0,   0,   0,    0,    0,    10,   990,  23760, 240240, 1261260, 3783780, 6726720, 7001280, 3938220, 923780;

The row polynomials satisfy n^2*P(n,x) = -((n^2-2*n-1)*x-(4*n+2)*x^2)*P(n-1,x) + ((2*n+1)*x^2+(2*n+1)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^2 - 2*n - 1)*x - (4*n + 2)*x^2)*P[n - 1, x] + ((2*n + 1)*x^2 + (2*n + 1)*x^3)*D[P[n - 1, x], x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-((n^2 - 2*n - 1)*x - (4*n + 2)*x^2)*v[i - 1] + ((2*n + 1)*x^2 + (2*n + 1)*x^3)*deriv(v[i - 1], x))/(n^2)); 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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • 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

Related ?

Same fingerprint ?

• A001263 → θ+1 → x ↦ x/(x+1)
• narayana → θ+1 → x ↦ x/(x+1)
• A008459 → x ↦ x+1 → D
• A253283 → D
• A069466 → basis A007318 → D
• A104684 → D

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.