A038329 → basis A013609

OEIS matches ?

• A009965: 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,            180,             36;
  1:          3375,           4050,           1620,            216;
  2:         50625,          81000,          48600,          12960,           1296;
  3:        759375,        1518750,        1215000,         486000,          97200,           7776;
  4:      11390625,       27337500,       27337500,       14580000,        4374000,         699840,          46656;
  5:     170859375,      478406250,      574087500,      382725000,      153090000,       36741600,        4898880,        279936;
  6:    2562890625,     8201250000,    11481750000,     9185400000,     4592700000,     1469664000,      293932800,      33592320,      1679616;
  7:   38443359375,   138396093750,   221433750000,   206671500000,   124002900000,    49601160000,    13226976000,    2267481600,    226748160,     10077696;
  8:  576650390625,  2306601562500,  4151882812500,  4428675000000,  3100072500000,  1488034800000,   496011600000,  113374080000,  17006112000,   1511654400,   60466176;
  9: 8649755859375, 38058925781250, 76117851562500, 91341421875000, 73073137500000, 40920957000000, 16368382800000, 4676680800000, 935336160000, 124711488000, 9976919040, 362797056;

The row polynomials satisfy P(n,x) = (15+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 180*x + 36*x^2.

P[0, x_] := 225 + 180*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 6*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 + 180*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 6*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-f3e6c91e6b1bdc2d
• 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-f3e6c91e6b1bdc2d
• 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-f3e6c91e6b1bdc2d

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 A013609 e-f3e6c91e6b1bdc2d

  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 (6)

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

• A013622 → basis A013611
• A038334 → basis A007318
• A038291 → basis A013611
• A013612 → basis A013610
• A013620 → x ↦ x/(x+1)
• A038220 → 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