A038320 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168695: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168743: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168791: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168839: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168887: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168983: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A169031: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168935: row sums, 14 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:          1452,           2310,           1224,             216;
  1:         15972,          34122,          27324,            9720,            1296;
  2:        175692,         471174,         505296,          270864,           72576,            7776;
  3:       1932612,        6237066,        8385300,         6011280,         2423520,          520992,          46656;
  4:      21258732,       80203398,      129660696,       116435880,        62726400,        20272032,        3639168,         279936;
  5:     233846052,     1009789770,     1907488044,      2058758856,      1388605680,       599350752,      161663040,       24914304,        1679616;
  6:    2572306572,    12510763782,    27041107104,     34091275680,     27627215616,     14924492352,     5374397952,     1244035584,      167961600,      10077696;
  7:   28295372292,   153052241034,   372516760836,    537250675104,    508447025856,    329932709568,   148665331584,    45930779136,     9311791104,    1118624256,     60466176;
  8:  311249095212,  1853346885126,  5015997815400,   8144857991160,   8816421335040,   6679941960384,  3614914904832,  1397230560000,   378014376960,   68175613440,   7376873472,   362797056;
  9: 3423740047332, 22254310307658, 66296057280156, 119689424795160, 145849782632400, 126377889574464, 79843715715456, 37059025588992, 12541541506560, 3018018009600, 490199288832, 48252008448, 2176782336;

The row polynomials satisfy P(n,x) = (11+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 1452 + 2310*x + 1224*x^2 + 216*x^3.

P[0, x_] := 1452 + 2310*x + 1224*x^2 + 216*x^3;
P[n_, x_] := P[n, x] = Expand[(11 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1452 + 2310*x + 1224*x^2 + 216*x^3; for(n = 1, N, my(i = n + 1); v[i] = (11 + 6*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-a06c6022a905308b

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-a06c6022a905308b

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-940e6d9217c9d73c root: theorem check

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

    • basis_map A013609 e-2fddb004b6f97d1c

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-da334b7c58401d78 root: theorem check

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

        • parent_provers periodic_product e-c5656743efdffc45 root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-de8595a50a8969d6 root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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.