A038331 → basis A007318

OEIS matches ?

• A009966: 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:            289,             170,              25;
  1:           4913,            4335,            1275,             125;
  2:          83521,           98260,           43350,            8500,            625;
  3:        1419857,         2088025,         1228250,          361250,          53125,           3125;
  4:       24137569,        42595710,        31320375,        12282500,        2709375,         318750,          15625;
  5:      410338673,       844814915,       745424925,       365404375,      107471875,       18965625,        1859375,         78125;
  6:     6975757441,     16413546920,     16896298300,      9938999000,     3654043750,      859775000,      126437500,      10625000,       390625;
  7:   118587876497,    313909084845,    369304805700,    253444474500,   111813738750,    32886393750,     6448312500,     812812500,     59765625,     1953125;
  8:  2015993900449,   5929393824850,   7847727121125,   6155080095000,  3168055931250,  1118137387500,   274053281250,   46059375000,   5080078125,   332031250,    9765625;
  9: 34271896307633, 110879664524695, 163058330183375, 143874997220625, 84632351306250, 34848615243750, 10249592718750, 2153275781250, 316658203125, 31044921875, 1826171875, 48828125;

The row polynomials satisfy P(n,x) = (17+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 170*x + 25*x^2.

P[0, x_] := 289 + 170*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 5*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 289 + 170*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 5*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-e0432c3ecb510b0f
• 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-e0432c3ecb510b0f
• 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-e0432c3ecb510b0f

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-e0432c3ecb510b0f

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-49423b91d6bf9513 root: theorem check

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

    • parent_provers periodic_product e-b8d4501ecbbc70e3 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