A075501 → 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:        252,           270,             37,              1;
  1:       1512,          3492,            936,             61,              1;
  2:       9072,         43416,          20340,           2400,             91,             1;
  3:      54432,        530064,         409536,          77940,           5130,           127,            1;
  4:     326592,       6415200,        7901712,        2280096,         231840,          9702,          169,           1;
  5:    1959552,      77308992,      148646016,       62624016,        9235296,        581112,        16800,         217,         1;
  6:   11757312,     929667456,     2752937280,     1651622400,      339682896,      30155328,      1286712,       27216,       271,       1;
  7:   70543872,   11167766784,    50482538496,    42391874880,    11842109280,    1425274704,     84197232,     2593080,     41850,     331,     1;
  8:  423263232,  134083745280,   919853459712,  1067887535616,   397655153280,   63151998624,   4961558448,   208665072,   4852980,   61710,   397,   1;
  9: 2539579392, 1609428206592, 16691446020096, 26549154314496, 12997542134016, 2671127103744, 271537453440, 14977481904, 470725992, 8555580, 87912, 469, 1;

The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 252 + 270*x + 37*x^2 + x^3.

P[0, x_] := 252 + 270*x + 37*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(6 + x)*P[n - 1, x] + 6*x*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] = 252 + 270*x + 37*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (6 + x)*v[i - 1] + 6*x*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-288a11b1cf9253fe

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-288a11b1cf9253fe

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 6).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-87ce0e9883ea7bf4 root: theorem check

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 6).
    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.