A038320 → basis A007318

OEIS matches ?

• A009967: 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,             204,              36;
  1:           4913,            5202,            1836,             216;
  2:          83521,          117912,           62424,           14688,            1296;
  3:        1419857,         2505630,         1768680,          624240,          110160,           7776;
  4:       24137569,        51114852,        45101340,        21224160,         5618160,         793152,          46656;
  5:      410338673,      1013777898,      1073411892,       631418760,       222853680,       47192544,        5552064,        279936;
  6:     6975757441,     19696256304,     24330669552,     17174590272,      7577025120,     2139395328,      377540352,      38071296,       1679616;
  7:   118587876497,    376690901814,    531798920208,    437952051936,    231856968672,    81831871296,    19254557952,    2912454144,     256981248,     10077696;
  8:  2015993900449,   7115272589820,  11300727054420,  10635978404160,   6569280779040,  2782283624064,   818318712960,  165039068160,   21843406080,   1713208320,    60466176;
  9: 34271896307633, 133055597429634, 234803995464060, 248615995197240, 175493643668640, 86714506283328, 30605119864704, 7715576436480, 1361572312320, 160184977920, 11307174912, 362797056;

The row polynomials satisfy P(n,x) = (17+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 204*x + 36*x^2.

P[0, x_] := 289 + 204*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 6*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 + 204*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 6*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-a19af80736e38760
• 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-a19af80736e38760
• 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-a19af80736e38760

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-a19af80736e38760

  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