A038305 → basis A007318

OEIS matches ?

• A001025: 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,            78,             9;
  1:          2197,          1521,           351,            27;
  2:         28561,         26364,          9126,          1404,            81;
  3:        371293,        428415,        197730,         45630,          5265,          243;
  4:       4826809,       6683274,       3855735,       1186380,        205335,        18954,          729;
  5:      62748517,     101362989,      70174377,      26990145,       6228495,       862407,        66339,        2187;
  6:     815730721,    1505964408,    1216355868,     561395016,     161940870,     29896776,      3449628,      227448,       6561;
  7:   10604499373,   22024729467,   20330519508,   10947202812,    3789416358,    874480698,    134535492,    13305708,     767637,     19683;
  8:  137858491849,  318134981190,  330370942005,  203305195080,   82104021090,  22736498148,   4372403490,   576580680,   49896405,   2558790,   59049;
  9: 1792160394037, 4549330231017, 5249227189635, 3634080362055, 1677267859410, 541886539194, 125050739814, 20612759310, 2378395305, 182953485, 8444007, 177147;

The row polynomials satisfy P(n,x) = (13+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 78*x + 9*x^2.

P[0, x_] := 169 + 78*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(13 + 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] = 169 + 78*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (13 + 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-ebc4517445c15609
• 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-ebc4517445c15609
• 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-ebc4517445c15609

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-ebc4517445c15609

  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 ?

Same fingerprint (1)

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

• A038327 → basis A013610

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