A201730: Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Matches ?

Row sums ? A006012

Submission text ?

Triangle begins:
   1:    1;
   2:    2;
   3:    5,     1;
   4:   14,     6;
   5:   41,    26,    1;
   6:  122,   100,   10;
   7:  365,   363,   63,   1;
   8: 1094,  1274,  322,  14;
   9: 3281,  4372, 1462, 116,  1;
  10: 9842, 14760, 6156, 744, 18;

The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (3-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.

P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (3 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = 4*v[i - 1] - (3 - x)*v[i - 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 Liu–Wang theorem for three-term recurrences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Liu–Wang theorem for three-term recurrences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Liu–Wang theorem for three-term recurrences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
  by Liu–Wang theorem for three-term recurrences
• Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 3]: every PF input has a real-rooted image with every root <= -c/b = 3
  by Constant-current basis map
• Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a PF image (zero allowed)
  by Constant-current basis map

Proof chains ? JSON

• Constant-current basis map [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ? step › direct check
• Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 10

• basis_map_A013611 of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0… Image under a basis map u^k -> P_k (A013611)
• shift_x_plus_one of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,… Image under a basis map u^k -> P_k (A007318)
• basis_map_A013609 of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0… Image under a basis map u^k -> P_k (A013609)
• basis_map_A013610 of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0… Image under a basis map u^k -> P_k (A013610)
• basis_map_A008277 of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0… Image under a basis map u^k -> P_k (A008277)
• one_plus_derivative of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0… Image under a real-rootedness preserver (derivative polynomial, shift 0)
• derivative of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,… Image under a real-rootedness preserver (derivative polynomial, shift 0)
• theta_plus_one_squared of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,… Image under a real-rootedness preserver (euler polynomial, shift 0)

Related ?

Same fingerprint ?

• A120910 → Veronese 2 → x ↦ x+1
• A372868 → x ↦ x+1
• A372873 → x ↦ x/(x+1)

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