A013612 → 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:        36,           85,            25;
  1:       216,         1115,           825,            125;
  2:      1296,        13345,         18775,           6750,            625;
  3:      7776,       153275,        367125,         235625,          50000,           3125;
  4:     46656,      1724905,       6640375,        6783750,        2478125,         346875,          15625;
  5:    279936,     19207235,     114870525,      175660625,       98350000,       23143750,        2296875,         78125;
  6:   1679616,    212679265,    1933964575,     4263225750,     3435403125,     1209206250,      198406250,      14687500,        390625;
  7:  10077696,   2347869995,   32006829525,    99197563625,   110636610000,    54662409375,    13188656250,    1594218750,      91406250,     1953125;
  8:  60466176,  25876958425,  523848622375,  2243182983750,  3372539678125,  2247717740625,   748103671875,  131306250000,   12175781250,   556640625,    9765625;
  9: 362797056, 284948873555, 8510962750125, 49726085770625, 98901946550000, 86541948350000, 38170320890625, 9124074609375, 1216617187500, 89267578125, 3330078125, 48828125;

The row polynomials satisfy P(n,x) = (6+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 36 + 85*x + 25*x^2.

P[0, x_] := 36 + 85*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 5*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 85*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 5*x)*v[i - 1] + 5*x*deriv(v[i - 1], 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-bab24e2df6037a14
• 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-bab24e2df6037a14
• 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-bab24e2df6037a14

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-bab24e2df6037a14

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

  • premise (sequence): Triangle of coefficients in expansion of (1+5x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-8bdb70660e92b5b3 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-41a50f5538a6b055 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [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
2. [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