A038321 → 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:            324,              301,                49;
  1:           5832,             9793,              3675,               343;
  2:         104976,           285649,            186151,             39102,              2401;
  3:        1889568,          7876057,           7956375,           2828035,            384160,             16807;
  4:       34012224,        210128401,         309736399,         165987990,          37467605,           3579891,           117649;
  5:      612220032,       5491295593,       11382463575,        8641686403,        2885425760,         452007458,         32118177,          823543;
  6:    11019960576,     141567930049,      402677903551,      416703014742,      193221389781,       44154375594,       5091142826,       280004620,        5764801;
  7:   198359290368,    3616337975257,    13876668423975,    19070162899795,    11805105033120,     3692731634949,     614549198718,     54398309322,     2386627614,      40353607;
  8:  3570467226624,   91796964414001,   469367755393999,   840873032059830,   676525971822085,   278350511884137,   62722073367723,   7946531115600,   557398608690,   19975035465,    282475249;
  9: 64268410079232, 2319917380936393, 15662346923505975, 36079622538091363, 37006305928234720, 19488258932613856, 5711777985252339, 971472098319261, 96873214852260, 5519768133495, 164683070167, 1977326743;

The row polynomials satisfy P(n,x) = (18+7*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 324 + 301*x + 49*x^2.

P[0, x_] := 324 + 301*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 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] = 324 + 301*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 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-86a79a465fd2222b
• 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-86a79a465fd2222b
• 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-86a79a465fd2222b

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-86a79a465fd2222b

  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)*11^(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-f6f8234ea600f6f8 root: theorem check

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

    • parent_provers periodic_product e-16634379058ff4dd 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