A038328 → basis A008277

OEIS matches ?

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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:           196,             60,              4;
  1:          2744,           1352,            192,              8;
  2:         38416,          27120,           6160,            544,            16;
  3:        537824,         510752,         165120,          23200,          1440,           32;
  4:       7529536,        9247680,        3993664,         794240,         78080,         3648,          64;
  5:     105413504,      163021952,       90381312,       23872128,       3306240,       243712,        8960,        128;
  6:    1475789056,     2819178240,     1952907520,      658205184,     120481536,     12461568,      720384,      21504,       256;
  7:   20661046784,    48058429952,    40790691840,    17069918720,    3967004160,    540040704,    43653120,    2042880,     50688,      512;
  8:  289254654976,   810256972800,   830349313024,   422979758080,  121413928960,  20894985216,  2215062528,  144506880,   5606400,   117760,   1024;
  9: 4049565169664, 13542620874752, 16566801580032, 10120293787648, 3517065953280, 744307503104, 99381596160, 8476317696, 457205760, 14981120, 270336, 2048;

The row polynomials satisfy P(n,x) = (14+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 196 + 60*x + 4*x^2.

P[0, x_] := 196 + 60*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 2*x)*P[n - 1, x] + 2*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] = 196 + 60*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 2*x)*v[i - 1] + 2*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-81c568f66e8f893e
• 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-81c568f66e8f893e
• 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-81c568f66e8f893e

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-81c568f66e8f893e

  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)*2^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ca869591900be157 root: theorem check

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

    • parent_provers periodic_product e-48411a1ad7cd0ada 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.

• A038255 → 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