A038329 → basis A007318

OEIS matches ?

• A001027: 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:           225,             90,              9;
  1:          3375,           2025,            405,             27;
  2:         50625,          40500,          12150,           1620,            81;
  3:        759375,         759375,         303750,          60750,          6075,           243;
  4:      11390625,       13668750,        6834375,        1822500,        273375,         21870,          729;
  5:     170859375,      239203125,      143521875,       47840625,       9568125,       1148175,        76545,        2187;
  6:    2562890625,     4100625000,     2870437500,     1148175000,     287043750,      45927000,      4592700,      262440,       6561;
  7:   38443359375,    69198046875,    55358437500,    25833937500,    7750181250,    1550036250,    206671500,    17714700,     885735,     19683;
  8:  576650390625,  1153300781250,  1037970703125,   553584375000,  193754531250,   46501087500,   7750181250,   885735000,   66430125,   2952450,   59049;
  9: 8649755859375, 19029462890625, 19029462890625, 11417677734375, 4567071093750, 1278779906250, 255755981250, 36536568750, 3653656875, 243577125, 9743085, 177147;

The row polynomials satisfy P(n,x) = (15+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 90*x + 9*x^2.

P[0, x_] := 225 + 90*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 90*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 3*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-fe430a680aa07591
• 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-fe430a680aa07591
• 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-fe430a680aa07591

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-fe430a680aa07591

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-d1ea7de7e1f61f8b root: theorem check

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

    • parent_provers periodic_product e-a6c3b06cf2d0ed0d root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (5)

Equal up to content, sign, reversal and a small index shift.

• A013611 → x ↦ x/(x+1)
• Triangle of coefficients in expansion of (1+5x)^n.
• A147716 → basis A013610
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
• A038291 → basis A013609

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