A038329 → 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:           225,             99,               9;
  1:          3375,           2457,             486,              27;
  2:         50625,          54351,           17577,            2106,             81;
  3:        759375,        1130193,          532170,          103275,           8505,            243;
  4:      11390625,       22621599,        14566149,         4075110,         539460,          32805,           729;
  5:     170859375,      441360657,       373753926,       141501087,       26790750,        2602530,        122472,         2187;
  6:    2562890625,     8457069951,      9172914417,      4517287866,     1147853511,      158448150,      11849166,       446148,        6561;
  7:   38443359375,   159915930993,    218002412610,    135933652035,    44543908395,     8197005033,     866366928,     51608826,     1594323,     19683;
  8:  576650390625,  2993816835999,   5057798457789,   3916414886670,  1610486482770,   379541876175,   53181123723,   4457018520,   217005075,   5609655,    59049;
  9: 8649755859375, 55618654219857, 115195218121566, 109167352653447, 55232379694800, 16217715733560, 2893602711384, 319996037889, 21834253485, 886620735, 19486170, 177147;

The row polynomials satisfy P(n,x) = (15+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 225 + 99*x + 9*x^2.

P[0, x_] := 225 + 99*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 3*x)*P[n - 1, x] + 3*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] = 225 + 99*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 3*x)*v[i - 1] + 3*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-f14e261f691e9869
• 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-f14e261f691e9869
• 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-f14e261f691e9869

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-f14e261f691e9869

  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)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-d1ea7de7e1f61f8b root: theorem check

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

    • parent_provers periodic_product e-a6c3b06cf2d0ed0d 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.

• A193685: 5-Stirling numbers of the second kind.

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