A038320 → basis A013611

OEIS matches ?

• A009985: 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,             816,              576;
  1:           4913,           20808,            29376,             13824;
  2:          83521,          471648,           998784,            940032,            331776;
  3:        1419857,        10022520,         28298880,          39951360,          28200960,           7962624;
  4:       24137569,       204459408,        721621440,        1358346240,        1438248960,         812187648,          191102976;
  5:      410338673,      4055111592,      17174590272,       40410800640,       57050542080,       48325165056,        22741254144,         4586471424;
  6:     6975757441,     78785025216,     389290712832,     1099173777408,     1939718430720,     2190740815872,      1546405281792,       623760113664,      110075314176;
  7:   118587876497,   1506763607256,    8508782723328,    28028931323904,    59355383980032,    83795836207104,     78866669371392,     47717648695296,    16841523068928,     2641807540224;
  8:  2015993900449,  28461090359280,  180811632870720,   680702617866240,  1681735879434240,  2849058431041536,   3351833448284160,   2704000092733440,  1431529460858880,   449107281838080,    63403380965376;
  9: 34271896307633, 532222389718536, 3756863927424960, 15911423692623360, 44926372779171840, 88795654434127872, 125358570965827584, 126412004335288320, 89232003060203520, 41991530851860480, 11856432240525312, 1521681143169024;

The row polynomials satisfy P(n,x) = (17+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 816*x + 576*x^2.

P[0, x_] := 289 + 816*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 24*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 + 816*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 24*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-5fcdd9bd1fb647dd
• 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-5fcdd9bd1fb647dd
• 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-5fcdd9bd1fb647dd

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 A013611 e-5fcdd9bd1fb647dd

  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