A038337 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168797: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168893: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168701: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168941: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167938: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168749: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168989: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168845: 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,            5016,            4477,             1331;
  1:         22464,           80784,          108900,            65219,            14641;
  2:        269568,         1216512,         2195424,          1980528,           893101,           161051;
  3:       3234816,        17563392,        39726720,         47916000,         32503020,         11756723,          1771561;
  4:      38817792,       246343680,       669917952,       1011985920,        917112240,        498613896,        150582685,         19487171;
  5:     465813504,      3383119872,     10748795904,      19512928512,      22137192000,      16071601392,       7291745076,       1890255587,        214358881;
  6:    5589762048,     45721387008,    166199869440,     352391897088,     480288517632,     436368328704,     264288556224,     102892262880,      23365118029,      2357947691;
  7:   67077144576,    610144026624,   2497333690368,    6056901328896,    9639773079552,   10519593638400,    7971514290432,    4141881273024,    1412196308028,    285311670611,     25937424601;
  8:  804925734912,   8059576909824,  36679588577280,  100153486540800,  182303191572480,  232272627535872,  211373701507584,  137389232471040,   62507049699600,  18957899435640,   3449677471933,   285311670611;
  9: 9659108818944, 105569106001920, 528810408935424, 1605317312839680, 3289326650818560, 4792606637727744, 5091483320985600, 3973781506235904, 2261366153576640, 915072339923280, 249933023455236, 41370192238595, 3138428376721;

The row polynomials satisfy P(n,x) = (12+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 5016*x + 4477*x^2 + 1331*x^3.

P[0, x_] := 1872 + 5016*x + 4477*x^2 + 1331*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 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] = 1872 + 5016*x + 4477*x^2 + 1331*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 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-11c598ddd1cff683

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-11c598ddd1cff683

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-37ab130d55b994ea root: theorem check

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

    • basis_map A007318 e-107a218257e6f127

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (1+11x)^n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-5d158048399db1f0 root: theorem check

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

        • parent_provers periodic_product e-70681fa17f5604ff root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-9aedabfb67a4638b root: theorem check

      Parent-row provers of polytool polytool ?

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