A038333 → 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:             361,              315,                49;
  1:            6859,            10717,              3822,               343;
  2:          130321,           326655,            201145,             40474,              2401;
  3:         2476099,          9405277,           8924370,           3026975,            396165,             16807;
  4:        47045881,        261869895,         360341149,         183549590,          39808580,           3680733,           117649;
  5:       893871739,       7137938437,       13724347182,        9864371643,        3155850390,         477419642,         32941720,           823543;
  6:     16983563041,     191843501535,      502869026065,      490645295994,      217375569831,       47871613398,       5351382414,        286592964,         5764801;
  7:    322687697779,    5106815981197,    17937582370890,    23145895022215,    13651168854015,     4106696112309,     661535621040,      56947998450,      2438510823,      40353607;
  8:   6131066257801,  135036029395575,   627687930107749,  1051398877484830,   803626201294210,   317319772042791,   69100545669603,    8503213241880,    581524300875,   20378571535,    282475249;
  9: 116490258898219, 3553854228089557, 21658953899324742, 46449770610147443, 45130223603221680, 22760651099370184, 6436371690145320, 1061922320135061, 103136815258785, 5741712971995, 167790297906, 1977326743;

The row polynomials satisfy P(n,x) = (19+7*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 361 + 315*x + 49*x^2.

P[0, x_] := 361 + 315*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 7*x)*P[n - 1, x] + 7*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] = 361 + 315*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 7*x)*v[i - 1] + 7*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-f4ff934f738cae69
• 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-f4ff934f738cae69
• 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-f4ff934f738cae69

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-f4ff934f738cae69

  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)*7^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-de9af033a7745ebd root: theorem check

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

    • parent_provers periodic_product e-400776ae4466efdf 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