A038333 → basis A007318

OEIS matches ?

• A009970: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,             266,              49;
  1:            6859,            7581,            2793,             343;
  2:          130321,          192052,          106134,           26068,            2401;
  3:         2476099,         4561235,         3360910,         1238230,          228095,           16807;
  4:        47045881,       103996158,        95785935,        47052740,        13001415,         1915998,          117649;
  5:       893871739,      2305248169,      2547905871,      1564503605,       576396065,       127413867,        15647317,         823543;
  6:     16983563041,     50056817384,     64546948732,     47560909592,     21903050470,      6455635928,      1189196092,      125178536,       5764801;
  7:    322687697779,   1069964471583,   1576789747596,   1355485923372,    749084326074,    275978435922,     67784177244,    10702764828,     985780971,     40353607;
  8:   6131066257801,  22588138844530,  37448756505405,  36791760777240,  23721003659010,  10487180565036,   3219748419090,   677841772440,   93649192245,   7667185330,   282475249;
  9: 116490258898219, 472092101850677, 869643345514405, 961184750305395, 708241394961870, 365303456348754, 134585483917962, 35417232609990, 6524227059735, 801220866985, 59037327041, 1977326743;

The row polynomials satisfy P(n,x) = (19+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 266*x + 49*x^2.

P[0, x_] := 361 + 266*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 7*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 266*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 7*x)*v[i - 1]); 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-ab827b425fb4aadf
• 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-ab827b425fb4aadf
• 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-ab827b425fb4aadf

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 A007318 e-ab827b425fb4aadf

  Geometric basis map (elementary) polytool ?

  • 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