A013618 → basis A013610

OEIS matches ?

• A009989: 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:          144,            792,            1089;
  1:         1728,          14256,           39204,            35937;
  2:        20736,         228096,          940896,          1724976,           1185921;
  3:       248832,        3421440,        18817920,         51749280,          71155260,          39135393;
  4:      2985984,       49268736,       338722560,       1241982720,        2561589360,        2817748296,         1291467969;
  5:     35831808,      689762304,      5690539008,      26081637120,       71724502080,      118345428432,       108483309396,        42618442977;
  6:    429981696,     9459597312,     91048624128,     500767432704,     1721388049920,     3787053709824,      5207198851008,      4091370525792,      1406408618241;
  7:   5159780352,   127704563712,   1404750200832,    9013813788672,    37181981878272,   102250450165248,    187459158636288,    220934008392768,    151892130770028,     46411484401953;
  8:  61917364224,  1702727516160,  21071253012480,  154522522091520,   743639637565440,  2454010803965952,   5623774759088640,   8837360335710720,   9113527846201680,   5569378128234360,   1531578985264449;
  9: 743008370688, 22476003213312, 309045044183040, 2549621614510080, 14022918879805440, 53988237687250944, 148467653639940096, 291632891078453760, 400995225232873920, 367578956463467760, 202168426054907268, 50542106513726817;

The row polynomials satisfy P(n,x) = (12+33*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 792*x + 1089*x^2.

P[0, x_] := 144 + 792*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 33*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 792*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 33*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-4217afbeac9f09d9
• 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-4217afbeac9f09d9
• 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-4217afbeac9f09d9

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-4217afbeac9f09d9

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of coefficients in expansion of (1+11x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-5d158048399db1f0 root: theorem check

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

    • parent_provers periodic_product e-70681fa17f5604ff 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.

• A038303 → basis A013611

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