A038335 → 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:           21,              9;
  2:          441,            459,             81;
  3:         9261,          17739,           7290,             729;
  4:       194481,         615519,         443961,          100602,           6561;
  5:      4084101,       20215899,       22854150,         8824545,        1279395,          59049;
  6:     85766121,      643233879,     1073254941,       629265510,      152346420,       15411789,        531441;
  7:   1801088541,    20068911459,    47646047610,     39864038949,    14347135530,     2388295854,     178564176,      4782969;
  8:  37822859361,   618277140639,  2038816059921,   2342288298042,  1176563075751,   286751746134,   34886975886,   2008846980,    43046721;
  9: 794280046581, 18888719953419, 85078320602670, 130779182845305, 88144690000185, 29514682926603, 5197288906656, 482725929294, 22082967873, 387420489;

The row polynomials satisfy P(n,x) = (21+9*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(21 + 9*x)*P[n - 1, x] + 9*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (21 + 9*x)*v[i - 1] + 9*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-2be01244d609a451
• 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-2be01244d609a451
• 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-2be01244d609a451

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-2be01244d609a451

  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)*12^(i-j)*9^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-03786f96284c3301 root: theorem check

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

    • parent_provers periodic_product e-2dcf274e8e5380c5 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A038284 → basis A008277

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