A395972 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(3 + 2\,x + 4\,x^{2} + 4\,n x + 8\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 18\,x + 20\,x^{2} + 24\,x^{3} + 20\,n x + 48\,n x^{2} + 64\,n x^{3} + 24\,n^{2} x^{2} + 56\,n^{2} x^{3} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-6\,x^{2} - 12\,x^{3} - 8\,x^{4} - 8\,n x^{3} - 16\,n x^{4} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 12 + 88\,x + 152\,x^{2} + 48\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	P_n(1)
1	12	88	152	48															300
2	24	328	1296	1584	384														3616
3	48	1056	7552	20736	19584	3840													52816
4	96	3104	35200	172800	364800	280320	46080												902400
5	192	8576	141696	1094400	4108800	7050240	4561920	645120											17610944
6	384	22656	514304	5757696	33868800	103111680	149022720	83220480	10321920										385840640
7	768	57856	1728512	26492928	225189888	1071759360	2745630720	3426877440	1682472960	185794560									9366004992
8	1536	143872	5474304	110149632	1278885888	8778792960	35115171840	77662126080	85279703040	37344706560	3715891200								249291046912
9	3072	350208	16537600	423014400	6436454400	60259368960	347250032640	1198374912000	2331721728000	2285273088000	902961561600	81749606400							7214466657280
10	6144	837632	48074752	1524019200	29439590400	360768798720	2832623861760	14073194741760	42714169344000	74187767808000	65644933939200	23625636249600	1961990553600						225432097824768
11	12288	1974272	135364608	5209915392	124597862400	1936589783040	19886175682560	134527152291840	587567121039360	1591664836608000	2496305981030400	2013002307993600	665114797670400	51011754393600					7561146661621760
12	24576	4595712	371032064	17045323776	494513750016	9508333486080	123749050613760	1094202131742720	6504423783137280	25349016052039680	62004787465420800	88641425217945600	65652127904563200	20047619476684800	1428329123020800				270855700473380864
13	49152	10584064	993886208	53727657984	1859302588416	43340310773760	697385000632320	7811387440496640	60667429304401920	321695426732359680	1131891316280524800	2523738532950835200	3314437729969766400	2269614976480051200	644176434482380800	42849873690624000			10317625746677612544
14	98304	24150016	2609971200	164015308800	6675254476800	185506950021120	3616945935482880	50066501861376000	492390959284224000	3410263310598144000	16323064245308620800	52348619440993075200	107231808910786560000	130227907791421440000	82914505591357440000	21981985203290112000	1371195958099968000		416355617207790469120
15	196608	54657024	6736052224	486958694400	23028616396800	752317460643840	17480440055070720	293084067287531520	3560884962066432000	31221234159058944000	195147302855874969600	851268921809751244800	2508054526960651468800	4751193994816389120000	5365946849364541440000	3192144190456725504000	793922459739881472000	46620662575398912000	17739392368047508750336

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Technical details

Node id: n-5482df9a9e2607d0 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-eddfbac374c8b398

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 2)
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:   12,     88,      152,         48;
   2:   24,    328,     1296,       1584,         384;
   3:   48,   1056,     7552,      20736,       19584,         3840;
   4:   96,   3104,    35200,     172800,      364800,       280320,         46080;
   5:  192,   8576,   141696,    1094400,     4108800,      7050240,       4561920,         645120;
   6:  384,  22656,   514304,    5757696,    33868800,    103111680,     149022720,       83220480,       10321920;
   7:  768,  57856,  1728512,   26492928,   225189888,   1071759360,    2745630720,     3426877440,     1682472960,      185794560;
   8: 1536, 143872,  5474304,  110149632,  1278885888,   8778792960,   35115171840,    77662126080,    85279703040,    37344706560,     3715891200;
   9: 3072, 350208, 16537600,  423014400,  6436454400,  60259368960,  347250032640,  1198374912000,  2331721728000,  2285273088000,   902961561600,    81749606400;
  10: 6144, 837632, 48074752, 1524019200, 29439590400, 360768798720, 2832623861760, 14073194741760, 42714169344000, 74187767808000, 65644933939200, 23625636249600, 1961990553600;

%F Formula

The row polynomials satisfy (3+(4*n+2)*x+(4*n^2+8*n+4)*x^2)*P(n,x) = (6+(20*n+18)*x+(24*n^2+48*n+20)*x^2+(16*n^3+56*n^2+64*n+24)*x^3)*P(n-1,x) - (6*x^2+(8*n+12)*x^3+(8*n^2+16*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 12 + 88*x + 152*x^2 + 48*x^3.

%t Mathematica

P[1, x_] := 12 + 88*x + 152*x^2 + 48*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((6 + (20*n + 18)*x + (24*n^2 + 48*n + 20)*x^2 + (16*n^3 + 56*n^2 + 64*n + 24)*x^3)*P[n - 1, x] - (6*x^2 + (8*n + 12)*x^3 + (8*n^2 + 16*n + 8)*x^4)*D[P[n - 1, x], x])/(3 + (4*n + 2)*x + (4*n^2 + 8*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 12 + 88*x + 152*x^2 + 48*x^3; for(n = 2, N, my(i = n); v[i] = ((6 + (20*n + 18)*x + (24*n^2 + 48*n + 20)*x^2 + (16*n^3 + 56*n^2 + 64*n + 24)*x^3)*v[i - 1] - (6*x^2 + (8*n + 12)*x^3 + (8*n^2 + 16*n + 8)*x^4)*deriv(v[i - 1], x))/(3 + (4*n + 2)*x + (4*n^2 + 8*n + 4)*x^2)); 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-f9379b86f8cefc86

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-f9379b86f8cefc86

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-b9d2d25190d8d499 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
  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
[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.