A038319 → 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:            256,             185,               25;
  1:           4096,            5165,             1575,              125;
  2:          65536,          128945,            66775,            11750,              625;
  3:        1048576,         3035525,          2380875,           698125,            81250,            3125;
  4:       16777216,        68988905,         77080375,         33546250,          6415625,          534375,           15625;
  5:      268435456,      1532653085,       2349034275,       1425335625,        398693750,        53987500,         3390625,          78125;
  6:     4294967296,     33527892065,      68738156575,      55930575750,      21479653125,      4206956250,       425906250,       20937500,        390625;
  7:    68719476736,    725560569845,    1954831531275,    2077538631125,    1052920391250,    279883471875,     40626468750,     3197343750,     126562500,      1953125;
  8:  1099511627776,  15580369350425,   54453422662375,   74177855221250,   48292827240625,  16739824303125,   3268234921875,   366196875000,   23074218750,    751953125,    9765625;
  9: 17592186044416, 332685314497805, 1493690835973875, 2571780625170625, 2109431056768750, 927796932631250, 234037927921875, 35017215234375, 3123140625000, 161240234375, 4404296875, 48828125;

The row polynomials satisfy P(n,x) = (16+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 256 + 185*x + 25*x^2.

P[0, x_] := 256 + 185*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 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] = 256 + 185*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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-82ec49a96db4a268
• 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-82ec49a96db4a268
• 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-82ec49a96db4a268

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-82ec49a96db4a268

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f6c78a0c1a54b2b2 root: theorem check

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

    • parent_provers periodic_product e-97cca34dfbf5cc39 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