A038333 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168937: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168841: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168697: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167931: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168985: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168889: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168745: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168793: 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,           3192,            1813,             343;
  1:         22464,          51408,           44100,           16807,            2401;
  2:        269568,         774144,          889056,          510384,          146461,           16807;
  3:       3234816,       11176704,        16087680,        12348000,         5330220,         1226911,          117649;
  4:      38817792,      156764160,       271289088,       260789760,       150398640,        52034472,        10000165,          823543;
  5:     465813504,     2152894464,      4352818176,      5028500736,      3630312000,      1677204144,       484243284,        79883671,        5764801;
  6:    5589762048,    29095428096,     67304079360,     90811736064,     78763249152,     45538633728,     17551348416,      4348307040,      628363309,       40353607;
  7:   67077144576,   388273471488,   1011316948992,   1560869388288,   1580841142272,   1097806348800,    529386617088,    175039123392,    37978508988,     4882786447,     282475249;
  8:  804925734912,  5128821669888,  14853717688320,  25809651302400,  29896179425280,  24239564181504,  14037283846656,   5806175800320,  1681015971600,   324443000280,   37569208117,   1977326743;
  9: 9659108818944, 67180340183040, 214146363949056, 413691839447040, 539421712220160, 500148026155008, 338124355430400, 167935096530432, 60815422261440, 15660427804560, 2721931499364, 286712377735, 13841287201;

The row polynomials satisfy P(n,x) = (12+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 3192*x + 1813*x^2 + 343*x^3.

P[0, x_] := 1872 + 3192*x + 1813*x^2 + 343*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 7*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 + 3192*x + 1813*x^2 + 343*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 7*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-946cc0d28b50f639

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-946cc0d28b50f639

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-de9af033a7745ebd root: theorem check

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

    • parent_provers periodic_product e-400776ae4466efdf root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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.