A075505 → 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:          1100,             730,               61,                1;
  1:         11000,           15700,             2560,              101,               1;
  2:        110000,          325000,            92500,             6600,             151,              1;
  3:       1100000,         6610000,          3100000,           356500,           14150,            211,             1;
  4:      11000000,       133300000,         99610000,         17360000,         1064000,          26810,           281,            1;
  5:     110000000,      2677000000,       3121600000,        794010000,        70560000,        2672600,         46480,          361,          1;
  6:    1100000000,     53650000000,      96325000000,      34882000000,      4322010000,      230916000,       5926200,        75360,        451,        1;
  7:   11000000000,   1074100000000,    2943400000000,    1491605000000,    250982500000,    18176970000,     645750000,     11955000,     115950,      551,      1;
  8:  110000000000,  21493000000000,   89376100000000,   62607600000000,  14040730000000,  1341600700000,   63379470000,   1602150000,   22390500,   171050,    661,   1;
  9: 1100000000000, 429970000000000, 2702776000000000, 2593680100000000, 764644100000000, 94536772000000, 5778163600000, 191551470000, 3617295000, 39495500, 243760, 781, 1;

The row polynomials satisfy P(n,x) = (10+x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1100 + 730*x + 61*x^2 + x^3.

P[0, x_] := 1100 + 730*x + 61*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(10 + x)*P[n - 1, x] + 10*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] = 1100 + 730*x + 61*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (10 + x)*v[i - 1] + 10*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-f660dc27e1b4f4f9

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-f660dc27e1b4f4f9

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 10).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-b1938fcbfe117d00 root: theorem check

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 10).
    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.