A200139 → basis A013610

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Submission text ?

Triangle begins:
   1:     1;
   2:     2,      3;
   3:     6,     15,      9;
   4:    18,     63,     72,      27;
   5:    54,    243,    405,     297,      81;
   6:   162,    891,   1944,    2106,    1134,     243;
   7:   486,   3159,   8505,   12150,    9720,    4131,     729;
   8:  1458,  10935,  34992,   61965,   65610,   41553,   14580,   2187;
   9:  4374,  37179, 137781,  290871,  382725,  321489,  168399,  50301,   6561;
  10: 13122, 124659, 524880, 1285956, 2020788, 2112642, 1469664, 656100, 170586, 19683;

The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.

P[1, x_] := 1;
P[2, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = ((3*n - 6 + (3*n - 6)*x)*v[i - 1])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 3
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operat…?
    • 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

Related ?

Same fingerprint ?

• A134239 → basis A013609
• A029653 → basis A013609
• A095660 → basis A013609
• A029600: Numbers in the (2,3)-Pascal triangle (by row).
• Veronese parent of A098172 → D
• A038208 → Pₙ + Pₙ₊₁
• A029618: Numbers in (3,2)-Pascal triangle (by row).

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.