A038301 → 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:             400,               561,                121;
  1:            8000,             21791,              11253,                1331;
  2:          160000,            763521,             712327,              194326,               14641;
  3:         3200000,          25429151,           38316465,            18134875,             3074610,              161051;
  4:        64000000,         823503681,         1889012191,          1382629490,           396258665,            45899535,            1771561;
  5:      1280000000,       26232614111,        88397052513,         94058497071,         40569478950,          7801310440,          657249131,          19487171;
  6:     25600000000,      827291037441,      4001234960767,       5957467922406,       3631090120581,       1031362551450,       142337840106,        9119996028,        214358881;
  7:    512000000000,    25927622160671,    177152069764065,     359759384455955,     297921914863650,     117294182685141,     23586042315066,     2450355855882,     123470715456,      2357947691;
  8:  10240000000000,   809388286980801,   7725590773858111,   21015920143570330,   23024355780289105,   12074204764885725,   3318635648632227,   497130983486280,   40288751683950,   1638773645245,    25937424601;
  9: 204800000000000, 25203676896404831, 333378083658829473, 1198825266121666711, 1704733891517776350, 1158833270949609530, 418218918196114497, 84726697533123657, 9819626000215680, 638190332307605, 21398375295825, 285311670611;

The row polynomials satisfy P(n,x) = (20+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 400 + 561*x + 121*x^2.

P[0, x_] := 400 + 561*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 11*x)*P[n - 1, x] + 11*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] = 400 + 561*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 11*x)*v[i - 1] + 11*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-f8d6f67754ace198
• 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-f8d6f67754ace198
• 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-f8d6f67754ace198

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-f8d6f67754ace198

  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)*9^(i-j)*11^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c85fc2223c79991e root: theorem check

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

    • parent_provers periodic_product e-38b35fa9e37eb25d 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