A112486 → 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:
  0:       2,        1;
  1:       3,        6,         3;
  2:       8,       31,        38,         15;
  3:      30,      180,       375,        330,         105;
  4:     144,     1198,      3644,       5215,        3570,         945;
  5:     840,     9072,     37044,      75432,       82215,       45990,       10395;
  6:    5760,    77292,    401624,    1079512,     1650320,     1451835,      686070,      135135;
  7:   45360,   732960,   4670100,   15786600,    31529190,    38553900,    28423395,    11621610,     2027025;
  8:  403200,  7663536,  58263552,  239433436,   596738120,   946655710,   964023060,   611485875,   220270050,   34459425;
  9: 3991680, 87626880, 778495608, 3792717456, 11422233108, 22487064600, 29559309780, 25780394640, 14346607275, 4617562950, 654729075;

The row polynomials satisfy P(n,x) = (n-1+(n-1)*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) + (n+(n+1)*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 3 + 6*x + 3*x^2.

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

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; v[2] = 3 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (n - 1)*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x) + (n + (n + 1)*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], 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-fa5fc5a793cceca0

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-fa5fc5a793cceca0

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

  • premise (sequence): Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check ma_wang_interval e-c24cc7aceb1eaafd root: theorem check

      Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

  • premise (sequence): Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
    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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.