A152842 → basis A008277

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:     1;
  1:     4,       3;
  2:     8,      13,        3;
  3:    32,     115,       69,        9;
  4:    64,     377,      391,      114,        9;
  5:   256,    2831,     5041,     2655,      486,       27;
  6:   512,    8749,    22995,    18316,     5571,      675,      27;
  7:  2048,   62779,   256197,   307093,   144084,    29538,    2619,     81;
  8:  4096,  190385,  1087567,  1791662,  1171597,   350850,   50490,   3348,    81;
  9: 16384, 1344983, 11446825, 26554307, 24120538, 10180941, 2163330, 235170, 12312, 243;

The row polynomials satisfy P(n,x) = (8+13*x+3*x^2)*P(n-2,x) + (13*x+6*x^2)*d/dx P(n-2,x) + 3*x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 13*x + 3*x^2)*P[n - 2, x] + (13*x + 6*x^2)*D[P[n - 2, x], x] + 3*x^2*D[P[n - 2, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 13*x + 3*x^2)*v[i - 2] + (13*x + 6*x^2)*deriv(v[i - 2], x) + 3*x^2*deriv(deriv(v[i - 2], x), x)); 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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-79898a409f2d103b
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-79898a409f2d103b
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-79898a409f2d103b

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A008277 e-79898a409f2d103b

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-596b986e3a00db44 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741