A038329 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168933: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A169029: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168693: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168789: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168981: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168837: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168885: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168741: row sums, 14 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:          1872,           1368,            333,             27;
  1:         22464,          22032,           8100,           1323,             81;
  2:        269568,         331776,         163296,          40176,           4941,           243;
  3:       3234816,        4790016,        2954880,         972000,         179820,         17739,           729;
  4:      38817792,       67184640,       49828608,       20528640,        5073840,        752328,         61965,         2187;
  5:     465813504,      922669056,      799497216,      395829504,      122472000,      24249456,       3000564,       212139,        6561;
  6:    5589762048,    12469469184,    12361973760,     7148445696,     2657152512,     658409472,     108755136,     11547360,      715149,      19683;
  7:   67077144576,   166402916352,   185752092672,   122867269632,    53331167232,   15872371200,    3280290048,    464833728,    43223868,    2381643,     59049;
  8:  804925734912,  2198066429952,  2728233861120,  2031663513600,  1008575815680,  350461956096,   86980594176,  15418874880,  1913187600,  158251320,   7853517,   177147;
  9: 9659108818944, 28791574364160, 39333005623296, 32564663746560, 18197900328960, 7231270920192, 2095152998400, 445968281088, 69214875840, 7638578640, 568996164, 25686315, 531441;

The row polynomials satisfy P(n,x) = (12+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 1368*x + 333*x^2 + 27*x^3.

P[0, x_] := 1872 + 1368*x + 333*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 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] = 1872 + 1368*x + 333*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-bebf949c785e986b

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

• obreschkoff_combination e-bebf949c785e986b

  Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

  • 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, +inf)
    • 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 ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
2. [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
3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.