Real-rooted atlas

A038328 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 912\,x + 148\,x^{2} + 8\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
0187291214882940
1224641468836003921641160
2269568221184725761190497632576240
3323481631933441313280288000355202336648067360
438817792447897602214604860825601002240990725440128112943040
5465813504615112704355332096117282816241920003193344263424124162561581202560
655897620488312979456549421056021180579845248696328670412895477766758402790451222136835840
7670771445761109352775688255648563236405116928105345515522090188800287981568272056321686528619521024309915701760
88049257349121465377619968121254838272060197437440019922485248046151368704763615641690243072074649600411648013619220484338819824640
996591088189441919438290944017481335832576964878925824035946469785609522661294081839366144002610148147227006566401986969609867264296960409660743477544960
1011590930582732824965081255116824816479580979215074814276403262433342259200186164875100164111771631616681091006464846108426247785676800515801088232980486430728192850408685629440
1113909116699279363227628362268672347727917481984023053073047879681050696392638464348264534638592865742345994241639663534080023775121244162626498068482176096665613111787525431296013844481638411905721598812160
12166909400391352324151336368707993648182606822375424346182460070952961721897132123750462805672009400321735419884470272369908093288448613234161745927906821931008786431213568592560783363274113024125239296296550432768166680102383370240
13200291280469622784531542244323229696661218009242664960511784165729894400275864147869040640109804749053755392333861730155233287909736888401920147569718067200021752869552128025250818424832228393536716815780151296080510976002860646406324224655362333521433367183360
14240349536563547340867790894928180019208997700599558438912746384600724406272043339381058882764801869385284383145984620243574293790720161689188691869696335278399448678405561738707599360738067212140544779088612556806461488889856412216197120195349708806480199681343488013107232669300067140567040

Computed in your browser ?

Technical details
Node id
n-4559e6a7432013d0 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-cd2a717ae5c77684

Observed on the stored rows ?

  • Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
  • 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:          1872,            912,            148,             8;
  1:         22464,          14688,           3600,           392,            16;
  2:        269568,         221184,          72576,         11904,           976,           32;
  3:       3234816,        3193344,        1313280,        288000,         35520,         2336,           64;
  4:      38817792,       44789760,       22146048,       6082560,       1002240,        99072,         5440,         128;
  5:     465813504,      615112704,      355332096,     117282816,      24192000,      3193344,       263424,       12416,        256;
  6:    5589762048,     8312979456,     5494210560,    2118057984,     524869632,     86704128,      9547776,      675840,      27904,       512;
  7:   67077144576,   110935277568,    82556485632,   36405116928,   10534551552,   2090188800,    287981568,    27205632,    1686528,     61952,    1024;
  8:  804925734912,  1465377619968,  1212548382720,  601974374400,  199224852480,  46151368704,   7636156416,   902430720,   74649600,   4116480,  136192,   2048;
  9: 9659108818944, 19194382909440, 17481335832576, 9648789258240, 3594646978560, 952266129408, 183936614400, 26101481472, 2700656640, 198696960, 9867264, 296960, 4096;
%F Formula
The row polynomials satisfy P(n,x) = (12+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 912*x + 148*x^2 + 8*x^3.
%t Mathematica
P[0, x_] := 1872 + 912*x + 148*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 2*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] = 1872 + 912*x + 148*x^2 + 8*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 2*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-9d869e2e35cfa66c

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.