A085880 → Pₙ + Pₙ₊₁

OEIS matches ?

No OEIS matches.

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:
   1:     2,      1;
   2:     3,      5,      2;
   3:     7,     19,     17,       5;
   4:    19,     71,     99,      61,      14;
   5:    56,    266,    504,     476,     224,      42;
   6:   174,   1002,   2400,    3060,    2190,     834,     132;
   7:   561,   3795,  10989,   17655,   16995,    9801,    3135,     429;
   8:  1859,  14443,  49049,   95095,  115115,   89089,   43043,   11869,   1430;
   9:  6292,  55198, 215072,  488488,  712712,  692692,  448448,  186472,  45188,   4862;
  10: 21658, 211718, 930852, 2423928, 4139772, 4845204, 3935568, 2190552, 799578, 172822, 16796;

The row polynomials satisfy (5*n^2-n-6+(4*n^2-2*n-6)*x)*P(n,x) = (20*n^2-34*n+6+(36*n^2-66*n+18)*x+(16*n^2-32*n+12)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 2 + x.

P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n^2 - 34*n + 6 + (36*n^2 - 66*n + 18)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 1, x])/(5*n^2 - n - 6 + (4*n^2 - 2*n - 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 2 + x; for(n = 2, N, my(i = n); v[i] = ((20*n^2 - 34*n + 6 + (36*n^2 - 66*n + 18)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 1])/(5*n^2 - n - 6 + (4*n^2 - 2*n - 6)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-4d30609870afaf16

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-4d30609870afaf16

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

  • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-68a5cf53bdad0f8f root: theorem check

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

    • preserver_image reversal, shift 0 e-93440eb5a87a1e1b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-68a5cf53bdad0f8f (expanded above)
  • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    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.