A038320 → basis A013610

OEIS matches ?

• A009979: 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,             612,              324;
  1:           4913,           15606,            16524,             5832;
  2:          83521,          353736,           561816,           396576,            104976;
  3:        1419857,         7516890,         15918120,         16854480,           8922960,           1889568;
  4:       24137569,       153344556,        405912060,        573052320,         455070960,         192735936,          34012224;
  5:      410338673,      3041333694,       9660707028,      17048306520,       18051148080,       11467788192,        4047454656,         612220032;
  6:     6975757441,     59088768912,     218976025968,     463713937344,      613739034720,      519873064704,      275226916608,       83261924352,      11019960576;
  7:   118587876497,   1130072705442,    4786190281872,   11824705402272,    18780414462432,    19885144724928,    14036572747008,     6369537212928,    1686053968128,     198359290368;
  8:  2015993900449,  21345817769460,  101706543489780,  287171416912320,   532111743102240,   676094920647552,   596554341747840,   360940442065920,  143314587290880,   33721079362560,   3570467226624;
  9: 34271896307633, 399166792288902, 2113235959176540, 6712631870325480, 14214985137159840, 21071625026848704, 22311132381369216, 16873965666581760, 8933275941131520, 3152920920399360, 667677371378688, 64268410079232;

The row polynomials satisfy P(n,x) = (17+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 612*x + 324*x^2.

P[0, x_] := 289 + 612*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 18*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 + 612*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 18*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-572c9115d911d6cf
• 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-572c9115d911d6cf
• 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-572c9115d911d6cf

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 A013610 e-572c9115d911d6cf

  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