Real-rooted atlas

A147716 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(14 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 15 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
015116
1210291240
229406164313600
34116011564121857154000
4576240203056286162016711810000
58067360341902460368056840301085112150000
611294304055933696118705441399440989804200991182250000
71581202560896014784222121312314627042785160157780558611312733750000
8221368358401412540953640057131526625991687045494449940802359847168127141006250000
930991570176021989256934470205393664132821015041648968384140372064829785633633689461411615093750000
10433881982464033884116725761202768080640256154814720363676588803614177280256542048130065604615801092015519226406250000
11607434775449605177658324070420227164801536478893548672076530203904086966140800720576595243863388819468680614460130901691138396093750000
128504086856294407856156429148163349568904622088727226161561615503164033280198282801024018784686412813346640384711195408280711207977201545618312075941406250000
1311905721598812160118490276864368645475012109385728155676855308083230431655808153643262756176640461268410803237469982950423303376096110419108839239200101410418018197131139121093750000
14166680102383370240177792109208928256884991972178370562726977185251737658172003662223369099951445544961078403336890889858481721088700947094848387620513281653539888534366561266356207762111467086816406250000

Computed in your browser ?

Technical details
Node id
n-eac767c2415c0225 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-f34fb4c93a9db674

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

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.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           15,            1;
  1:          210,           29,           1;
  2:         2940,          616,          43,           1;
  3:        41160,        11564,        1218,          57,          1;
  4:       576240,       203056,       28616,        2016,         71,         1;
  5:      8067360,      3419024,      603680,       56840,       3010,        85,       1;
  6:    112943040,     55933696,    11870544,     1399440,      98980,      4200,      99,      1;
  7:   1581202560,    896014784,   222121312,    31462704,    2785160,    157780,    5586,    113,    1;
  8:  22136835840,  14125409536,  4005713152,   662599168,   70454944,   4994080,  235984,   7168,  127,   1;
  9: 309915701760, 219892569344, 70205393664, 13282101504, 1648968384, 140372064, 8297856, 336336, 8946, 141, 1;
%F Formula
The row polynomials satisfy P(n,x) = (14+x)*P(n-1,x) for n >= 1, with P(0,x) = 15 + x.
%t Mathematica
P[0, x_] := 15 + x;
P[n_, x_] := P[n, x] = Expand[(14 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 15 + x; for(n = 1, N, my(i = n + 1); v[i] = (14 + x)*v[i - 1]); 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)

Generated with polytool c48629374597.

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-82f056fc366df894

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

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.