A143491 → Pₙ + Pₙ₊₁

OEIS matches ?

• A171005: 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:         30,          31,          10,          1;
  1:        144,         180,          80,         15,          1;
  2:        840,        1198,         651,        169,         21,         1;
  3:       5760,        9072,        5684,       1820,        315,        28,         1;
  4:      45360,       77292,       53964,      20089,       4320,       538,        36,        1;
  5:     403200,      732960,      557864,     233100,      58674,      9135,       861,       45,       1;
  6:    3991680,     7663536,     6262740,    2870440,     815815,    149513,     17710,     1310,      55,     1;
  7:   43545600,    87626880,    76044672,   37622904,   11789756,   2451834,    343497,    32076,    1914,    66,    1;
  8:  518918400,  1088068320,   994409208,  524863756,  178338810,  41070875,   6556914,   727143,   54990,  2705,   78,  1;
  9: 6706022400, 14583179520, 13945097088, 7782670896, 2832239696, 709949240, 125984144, 15978963, 1440153, 90090, 3718, 91, 1;

The row polynomials satisfy (n+4+x)*P(n,x) = (n^2+8*n+15+(2*n+8)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 30 + 31*x + 10*x^2 + x^3.

P[0, x_] := 30 + 31*x + 10*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 8*n + 15 + (2*n + 8)*x + x^2)*P[n - 1, x])/(n + 4 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 30 + 31*x + 10*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 8*n + 15 + (2*n + 8)*x + x^2)*v[i - 1])/(n + 4 + x)); 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-a410505258437c48

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-a410505258437c48

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

  • premise (sequence): Unsigned 2-Stirling numbers of the first kind.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-84f3431e657be6ed root: theorem check

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

    • parent_provers root_domain e-25f97869157cba20 root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Unsigned 2-Stirling numbers of the first kind.
    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.