A038320 → basis A013609

OEIS matches ?

• A009973: 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,             408,             144;
  1:           4913,           10404,            7344,             1728;
  2:          83521,          235824,          249696,           117504,            20736;
  3:        1419857,         5011260,         7074720,          4993920,          1762560,           248832;
  4:       24137569,       102229704,       180405360,        169793280,         89890560,         25380864,          2985984;
  5:      410338673,      2027555796,      4293647568,       5051350080,       3565658880,       1510161408,        355332096,        35831808;
  6:     6975757441,     39392512608,     97322678208,     137396722176,     121232401920,      68460650496,      24162582528,      4873125888,       429981696;
  7:   118587876497,    753381803628,   2127195680832,    3503616415488,    3709711498752,    2618619881472,    1232291708928,    372794130432,     65787199488,     5159780352;
  8:  2015993900449,  14230545179640,  45202908217680,   85087827233280,  105108492464640,   89033075970048,   52372397629440,  21125000724480,   5591911956480,   877162659840,    61917364224;
  9: 34271896307633, 266111194859268, 939215981856240, 1988927961577920, 2807898298698240, 2774864201066496, 1958727671341056, 987593783869440, 348562511953920, 82014708695040, 11578547109888, 743008370688;

The row polynomials satisfy P(n,x) = (17+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 408*x + 144*x^2.

P[0, x_] := 289 + 408*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 12*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 + 408*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 12*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-7304be3f66cc3caf
• 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-7304be3f66cc3caf
• 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-7304be3f66cc3caf

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 A013609 e-7304be3f66cc3caf

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-940e6d9217c9d73c root: theorem check

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

    • basis_map A013609 e-2fddb004b6f97d1c

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-da334b7c58401d78 root: theorem check

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

        • parent_provers periodic_product e-c5656743efdffc45 root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-de8595a50a8969d6 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