A027467 → Pₙ + Pₙ₊₁

OEIS matches ?

• A167926: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168982: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168790: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168838: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168742: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168886: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168934: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168694: 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:           16,            1;
  1:          240,           31,            1;
  2:         3600,          705,           46,           1;
  3:        54000,        14175,         1395,          61,          1;
  4:       810000,       266625,        35100,        2310,         76,         1;
  5:     12150000,      4809375,       793125,       69750,       3450,        91,        1;
  6:    182250000,     84290625,     16706250,     1839375,     121500,      4815,      106,      1;
  7:   2733750000,   1446609375,    334884375,    44296875,    3661875,    193725,     6405,    121,     1;
  8:  41006250000,  24432890625,   6469875000,   999337500,   99225000,   6567750,   289800,   8220,   136,   1;
  9: 615093750000, 407499609375, 121481015625, 21459937500, 2487712500, 197741250, 10914750, 413100, 10260, 151, 1;

The row polynomials satisfy P(n,x) = (15+x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + x.

P[0, x_] := 16 + x;
P[n_, x_] := P[n, x] = Expand[(15 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 16 + x; for(n = 1, N, my(i = n + 1); v[i] = (15 + 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-e331e686933fdde7

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-e331e686933fdde7

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

  • premise (sequence): Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-14b8bc50df3e26b6 root: theorem check

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

    • parent_provers periodic_product e-5234407c4fbd8679 root: theorem check

      Parent-row provers of polytool polytool ?

    • basis_map A007318 e-7eb5590197b773bf

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (14 + x)^n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-08b3ad2096e0d7df root: theorem check

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

        • parent_provers periodic_product e-7403f1e58efb6a95 root: theorem check

          Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
    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.