A038301 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168890: 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
• A168938: 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
• A168746: 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
• A167933: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168794: 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:          810,          2871,           3388,            1331;
  1:         7290,         34749,          62073,           49247,           14641;
  2:        65610,        392931,         940896,         1126026,          673486,          161051;
  3:       590490,       4258089,       12790305,        20484090,        18447660,         8857805,         1771561;
  4:      5314410,      44818191,      161951724,       325050165,       391353930,       282644505,       113379904,        19487171;
  5:     47829690,     461822229,     1950565617,      4706920449,      7097737185,      6848693775,      4129508691,      1422563483,       214358881;
  6:    430467210,    4682526651,    22635135072,     63818505828,    115655759604,    139713353010,    112501209744,     58227666948,     17577428242,      2357947691;
  7:   3874204890,   46877879169,   255224008809,    823353038244,   1742905400544,   2529633532734,   2549357770806,   1761562309716,    798701190606,    214573239881,     25937424601;
  8:  34867844010,  464517166311,  2812672750140,  10217641441095,  24743032025580,  41938661200590,  50770188797328,  43896996266310,  26565496122330,  10716872255595,   2593742460100,   285311670611;
  9: 313810596090, 4564200780909, 30423743580681, 122898173221395, 335081344082265, 649621303086690, 918256972382442, 953545043167398, 721956424030380, 388672307645985, 141229276952445, 31098972096599, 3138428376721;

The row polynomials satisfy P(n,x) = (9+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 810 + 2871*x + 3388*x^2 + 1331*x^3.

P[0, x_] := 810 + 2871*x + 3388*x^2 + 1331*x^3;
P[n_, x_] := P[n, x] = Expand[(9 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 810 + 2871*x + 3388*x^2 + 1331*x^3; for(n = 1, N, my(i = n + 1); v[i] = (9 + 11*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-0ccb5845893f0f92

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-0ccb5845893f0f92

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-c85fc2223c79991e root: theorem check

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

    • parent_provers periodic_product e-38b35fa9e37eb25d root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^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.