A038305 → basis A013610

OEIS matches ?

• A009966: 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:           169,            234,             81;
  1:          2197,           4563,           3159,            729;
  2:         28561,          79092,          82134,          37908,            6561;
  3:        371293,        1285245,        1779570,        1232010,          426465,           59049;
  4:       4826809,       20049822,       34701615,       32032260,        16632135,         4605822,         531441;
  5:      62748517,      304088967,      631569393,      728733915,       504508095,       209564901,       48361131,        4782969;
  6:     815730721,     4517893224,    10947202812,    15157665432,     13117210470,      7264916568,     2514778812,      497428776,       43046721;
  7:   10604499373,    66074188401,   182974675572,   295574475924,    306942724998,    212498809614,    98076373668,    29099583396,     5036466357,     387420489;
  8:  137858491849,   954404943570,  2973338478045,  5489240267160,   6650425708290,   5524969049964,  3187482144210,  1260981947160,   327370313205,   50364663570,   3486784401;
  9: 1792160394037, 13647990693051, 47243044706715, 98120169775485, 135858696612210, 131678429024142, 91161989324406, 45080104610970, 15604651596105, 3601073445255, 498610169343, 31381059609;

The row polynomials satisfy P(n,x) = (13+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 234*x + 81*x^2.

P[0, x_] := 169 + 234*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(13 + 9*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 234*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (13 + 9*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-c80be26a429cc3fc
• 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-c80be26a429cc3fc
• 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-c80be26a429cc3fc

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-c80be26a429cc3fc

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*10^(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-e186153c671b61f0 root: theorem check

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

    • parent_provers periodic_product e-f733aee5869de636 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