A038334 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168794: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168938: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168746: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167933: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168890: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168842: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168698: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168986: row sums, 13 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,           3648,            2368,             512;
  1:         22464,          58752,           57600,           25088,            4096;
  2:        269568,         884736,         1161216,          761856,          249856,           32768;
  3:       3234816,       12773376,        21012480,        18432000,         9093120,         2392064,          262144;
  4:      38817792,      179159040,       354336768,       389283840,       256573440,       101449728,        22282240,         2097152;
  5:     465813504,     2460450816,      5685313536,      7506100224,      6193152000,      3269984256,      1078984704,       203423744,        16777216;
  6:    5589762048,    33251917824,     87907368960,    135555710976,    134366625792,     88785027072,     39107690496,     11072962560,      1828716544,      134217728;
  7:   67077144576,   443741110272,   1320903770112,   2329927483392,   2696845197312,   2140353331200,   1179572502528,    445737074688,    110528299008,    16240345088,     1073741824;
  8:  804925734912,  5861510479872,  19400774123520,  38526359961600,  51001562234880,  47259001552896,  31277696679936,  14785424916480,   4892236185600,  1079110533120,   142807662592,    8589934592;
  9: 9659108818944, 76777531637760, 279701373321216, 617522512527360, 920229626511360, 975120516513792, 753404372582400, 427646672437248, 176990233559040, 52087215882240, 10346576216064, 1245540515840, 68719476736;

The row polynomials satisfy P(n,x) = (12+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 3648*x + 2368*x^2 + 512*x^3.

P[0, x_] := 1872 + 3648*x + 2368*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 8*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 + 3648*x + 2368*x^2 + 512*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 8*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-acb1eb270a760ac4

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-acb1eb270a760ac4

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-f4f0736bd14a35db root: theorem check

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

    • parent_provers periodic_product e-b445a5ff4f115ce0 root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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.