A075503 → 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:         576,            472,              49,               1;
  1:        4608,           8128,            1648,              81,               1;
  2:       36864,         134656,           47680,            4240,             121,              1;
  3:      294912,        2191360,         1278976,          183360,            9080,            169,             1;
  4:     2359296,       35356672,        32886784,         7146496,          546560,          17192,           225,           1;
  5:    18874368,      568066048,       824639488,       261574656,        29008896,        1371776,         29792,         289,          1;
  6:   150994944,     9107931136,     20359413760,      9195028480,      1421930496,       94854144,       3040128,       48288,        361,        1;
  7:  1207959552,   145877893120,    497733861376,    314600325120,     66072248320,     5974929408,     265101312,     6130560,      74280,      441,      1;
  8:  9663676416,  2335254249472,  12091490566144,  10564944265216,   2957490257920,   352868859904,   20820602880,   657457152,   11478720,   109560,    529,   1;
  9: 77309411328, 37373731667968, 292531027836928, 350169707053056, 128864554582016, 19895195533312, 1518822621184, 62897860608, 1483924992, 20243520, 156112, 625, 1;

The row polynomials satisfy P(n,x) = (8+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 576 + 472*x + 49*x^2 + x^3.

P[0, x_] := 576 + 472*x + 49*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(8 + x)*P[n - 1, x] + 8*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] = 576 + 472*x + 49*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (8 + x)*v[i - 1] + 8*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-2d4f988e0d40946e

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-2d4f988e0d40946e

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 8).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-c4869f0cf2b30feb root: theorem check

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 8).
    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.