A225472 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	3	3															6
2	6	24	18														48
3	12	138	288	162													600
4	24	726	3132	4374	1944												10200
5	48	3702	29412	76302	79704	29160											218328
6	96	18654	257508	1104030	2031480	1691280	524880										5627928
7	192	93558	2171988	14461902	41689080	59223960	40940640	11022480									169603800
8	384	468366	17937252	178628814	757189944	1632143520	1884844080	1113270480	264539520								5849022360
9	768	2342982	146308212	2126352222	12744204984	39104289000	67075464960	65186946720	33596519040	7142567040							227124995928
10	1536	11717214	1184523588	24706648350	203935096440	855935987760	2045386501200	2907884538720	2437996216320	1114240458240	214277011200						9805558700568
11	3072	58590678	9546491988	282433844142	3151571130360	17609938238520	56314577803680	109834460915760	133177130553600	98138871129600	40284078105600	7071141369600					465873808176600
12	6144	292962606	76723479972	3192690713454	47511201954744	346642517010240	1443270444366960	3708798734939760	6098632456371840	6441809787705600	4233256633267200	1576864525420800	254561089305600				24154617097504920
13	12288	1464831462	615545615412	35810109167742	703469115927864	6605590818495240	35104974193523520	115611050235320640	247575613504222080	351475560165502080	328718505895718400	194887727287545600	66440444308761600	9927882482918400			1357087245127562328
14	24576	7324194174	4933153912068	399451111383870	10278288933003000	122847080653337040	821000118603384720	3396258613476368640	9211631156757469440	16877332809413556480	21063258993628051200	17668781149622803200	9540695066084582400	2998220509841356800	416971064282572800		82127679242886000408
15	49152	36621044598	39509176461588	4438360610431182	148689458398648440	2242574705101774680	18631249823827761120	95354950600627557840	321012616799127052800	738156692705444812800	1180344272078504332800	1313494887026523801600	998622533897635046400	495014148480794342400	144271988241770188800	18763697892715776000	5326062779616879082200

Computed in your browser ?

Technical details

Node id: n-464957f72336fbd0 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-24e672df326049ba

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 0)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    3,        3;
   2:    6,       24,         18;
   3:   12,      138,        288,         162;
   4:   24,      726,       3132,        4374,         1944;
   5:   48,     3702,      29412,       76302,        79704,        29160;
   6:   96,    18654,     257508,     1104030,      2031480,      1691280,        524880;
   7:  192,    93558,    2171988,    14461902,     41689080,     59223960,      40940640,      11022480;
   8:  384,   468366,   17937252,   178628814,    757189944,   1632143520,    1884844080,    1113270480,     264539520;
   9:  768,  2342982,  146308212,  2126352222,  12744204984,  39104289000,   67075464960,   65186946720,   33596519040,    7142567040;
  10: 1536, 11717214, 1184523588, 24706648350, 203935096440, 855935987760, 2045386501200, 2907884538720, 2437996216320, 1114240458240, 214277011200;

%F Formula

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

%t Mathematica

P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x] + (3*x + 3*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 3 + 3*x; for(n = 2, N, my(i = n); v[i] = (2 + 3*x)*v[i - 1] + (3*x + 3*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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-2dc56bf9965267b7

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-2dc56bf9965267b7

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-2d30e6aff83a7d44 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.