A038324 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168939: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168891: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168699: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167935: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168747: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168843: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168795: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168987: 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:          12,           10;
  1:         132,          230,           100;
  2:        1452,         3850,          3400,          1000;
  3:       15972,        56870,         75900,         45000,         10000;
  4:      175692,       785290,       1403600,       1254000,        560000,        100000;
  5:     1932612,     10395110,      23292500,      27830000,      18700000,       6700000,       1000000;
  6:    21258732,    133672330,     360168600,     539055000,     484000000,     260700000,      78000000,      10000000;
  7:   233846052,   1682982950,    5298577900,    9531291000,   10714550000,    7707700000,    3465000000,     890000000,    100000000;
  8:  2572306572,  20851272970,   75114186400,  157829980000,  213172960000,  191930200000,  115192000000,   44440000000,  10000000000,   1000000000;
  9: 28295372292, 255087068390, 1034768780100, 2487271644000, 3923202360000, 4242961800000, 3186414000000, 1640760000000, 554400000000, 111000000000, 10000000000;

The row polynomials satisfy P(n,x) = (11+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 10*x.

P[0, x_] := 12 + 10*x;
P[n_, x_] := P[n, x] = Expand[(11 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 10*x; for(n = 1, N, my(i = n + 1); v[i] = (11 + 10*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-4af761c482728828

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-4af761c482728828

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

  • premise (sequence): Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-0ddfaa48a1d96c8a root: theorem check

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

    • basis_map A013609 e-0f31cc897fc45a50

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-0a7984b7b9125974 root: theorem check

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

        • parent_provers periodic_product e-7949976749bfedcf root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-96e2fb6c9c1a89f9 root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^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.