A219512 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 - 3\,x + 3\,n x\right)\,P_{n-2}(x) + \left(x - x^{2}\right)\,P_{n-2}'(x) + \left(3\,x + 3\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2 + 3\,x\)

\(P_{1}(x) = 2 + 15\,x + 15\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	3															5
1	2	15	15														32
2	2	45	156	105													308
3	2	111	927	1935	945												3920
4	2	249	4236	19368	27630	10395											61880
5	2	531	16707	142890	423180	446355	135135										1164800
6	2	1101	60228	878643	4635306	9792135	8056440	2027025									25450880
7	2	2247	204927	4799817	41251977	150269445	241137765	160826715	34459425								632952320
8	2	4545	670980	24181530	319195620	1837695708	4965823620	6326742870	3520588050	654729075							17649632000
9	2	9147	2140227	115039692	2236739832	19214220870	80280418230	169094060460	176723979390	83896849575	13749310575						545312768000
10	2	18357	6704268	524966733	14574484908	179304232986	1094870764656	3506125954410	5968457556930	5246609594625	2162802618900	316234143225					18489511040000
11	2	36783	20736975	2322930435	89911612665	1536345242838	13169974259862	60668487141750	154904992338420	219063193987995	165199561501275	59986429878375	7905853580625				682527093248000
12	2	73641	63571932	10043758620	531893372850	12327118117713	143940197691864	916899078848688	3325506920070390	6975301043252475	8373600893731500	5504177282152500	1781447430712950	213458046676875			27247200012032000
13	2	147363	193661475	42670932030	3045494845260	93942781819173	1459561078853577	12480587793777780	61849054178423730	182372916397021245	321652844778836565	333516204460659150	193609301758790400	56408118014122875	6190283353629375		1169635902955520000
14	2	294813	587321076	178867435479	16995200700870	687158174507553	13941781145733672	156372215786354979	1028325102394737006	4102614977129576055	10083322079209751220	15235412369630573925	13840138353267434250	7173499985506406475	1897273652616342000	191898783962510625	53723503633479680000

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Technical details

Node id: n-f055c1e733379fba · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-f985e5e5089e1527

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): no

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.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2,    3;
  1: 2,   15,      15;
  2: 2,   45,     156,       105;
  3: 2,  111,     927,      1935,        945;
  4: 2,  249,    4236,     19368,      27630,       10395;
  5: 2,  531,   16707,    142890,     423180,      446355,      135135;
  6: 2, 1101,   60228,    878643,    4635306,     9792135,     8056440,      2027025;
  7: 2, 2247,  204927,   4799817,   41251977,   150269445,   241137765,    160826715,     34459425;
  8: 2, 4545,  670980,  24181530,  319195620,  1837695708,  4965823620,   6326742870,   3520588050,   654729075;
  9: 2, 9147, 2140227, 115039692, 2236739832, 19214220870, 80280418230, 169094060460, 176723979390, 83896849575, 13749310575;

%F Formula

The row polynomials satisfy P(n,x) = (1+(3*n-3)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (3*n+3)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 3*x, P(1,x) = 2 + 15*x + 15*x^2.

%t Mathematica

P[0, x_] := 2 + 3*x;
P[1, x_] := 2 + 15*x + 15*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 3)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (3*n + 3)*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 3*x; v[2] = 2 + 15*x + 15*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (3*n - 3)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (3*n + 3)*x*v[i - 1] + (x - x^2)*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)

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-b4f1d8e4113d9eff

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-b4f1d8e4113d9eff

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-5be8dcded4b783ba root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.
  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

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[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
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.