A038329 → basis A013610

OEIS matches ?

• A009968: 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,            270,              81;
  1:          3375,           6075,            3645,             729;
  2:         50625,         121500,          109350,           43740,            6561;
  3:        759375,        2278125,         2733750,         1640250,          492075,           59049;
  4:      11390625,       41006250,        61509375,        49207500,        22143375,         5314410,          531441;
  5:     170859375,      717609375,      1291696875,      1291696875,       775018125,       279006525,        55801305,        4782969;
  6:    2562890625,    12301875000,     25833937500,     31000725000,     23250543750,     11160261000,      3348078300,      573956280,       43046721;
  7:   38443359375,   207594140625,    498225937500,    697516312500,    627764681250,    376658808750,    150663523500,    38742048900,     5811307335,     387420489;
  8:  576650390625,  3459902343750,   9341736328125,  14946778125000,  15694117031250,  11299764262500,   5649882131250,  1937102445000,   435848050125,   58113073350,   3486784401;
  9: 8649755859375, 57088388671875, 171265166015625, 308277298828125, 369932758593750, 310743517218750, 186446110331250, 79905475856250, 23971642756875, 4794328551375, 575319426165, 31381059609;

The row polynomials satisfy P(n,x) = (15+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 270*x + 81*x^2.

P[0, x_] := 225 + 270*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 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] = 225 + 270*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 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-a617f0e181e2ec6f
• 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-a617f0e181e2ec6f
• 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-a617f0e181e2ec6f

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-a617f0e181e2ec6f

  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 (4)

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

• A038271 → basis A013611
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
• A013612 → basis A013609
• A013620 → basis A007318

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