A225467 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(3 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	4	4															8
2	12	44	16														72
3	36	356	352	64													808
4	108	2636	5296	2368	256												10664
5	324	18884	68800	56704	14336	1024											160072
6	972	133484	832336	1125760	499200	80896	4096										2676744
7	2916	938276	9689632	20215744	13987840	3857408	434176	16384									49142376
8	8748	6579596	110339056	341994688	346631936	144671744	27152384	2244608	65536								979688296
9	26244	46092164	1240048000	5571276544	7953985536	4713977856	1311801344	178192384	11272192	262144							21026934408
10	78732	322750124	13824896656	88529340160	173410831360	140237432832	54274547712	10771169280	1107296256	55312384	1048576						482534704072
11	236196	2259565796	153364863712	1383239689024	3648923156480	3919104280576	2026362519552	551004438528	81840046080	6586368000	266338304	4194304					11772955696552
12	708588	15817905356	1696051764016	21362054790208	74862498729216	104735091079168	70388205150208	25186587672576	5068419366912	584228536320	37798019072	1262485504	16777216				303938032984360
13	2125764	110728171844	18719841025600	327215028909184	1507835695015936	2708357089737728	2319421903372288	1062337038450688	278141028532224	43058590384128	3962228965376	210528894976	5905580032	67108864			8269375676274632
14	6377292	775105705964	206361163968976	4983104797740160	29957738320939520	68323555844031488	73457819750002688	42210135805460480	13984284152430592	2791849139109888	342610207047680	25743773925376	1143300161536	27313307648	268435456		236285148948644744
15	19131876	5425765450916	2273073226482592	75572016621978304	589129447288811520	1691272737696482304	2256655356626198528	1602345488969285632	658290488556912640	164819253035008000	25899635459489792	2580398202683392	161283403939840	6075432566784	125090922496	1073741824	7069010806469086440

Computed in your browser ?

Technical details

Node id: n-54cd774d491372da · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-256de4c419ed5564

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:     4,         4;
   2:    12,        44,          16;
   3:    36,       356,         352,          64;
   4:   108,      2636,        5296,        2368,          256;
   5:   324,     18884,       68800,       56704,        14336,         1024;
   6:   972,    133484,      832336,     1125760,       499200,        80896,        4096;
   7:  2916,    938276,     9689632,    20215744,     13987840,      3857408,      434176,       16384;
   8:  8748,   6579596,   110339056,   341994688,    346631936,    144671744,    27152384,     2244608,      65536;
   9: 26244,  46092164,  1240048000,  5571276544,   7953985536,   4713977856,  1311801344,   178192384,   11272192,   262144;
  10: 78732, 322750124, 13824896656, 88529340160, 173410831360, 140237432832, 54274547712, 10771169280, 1107296256, 55312384, 1048576;

%F Formula

The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 4*x.

%t Mathematica

P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(3 + 4*x)*P[n - 1, x] + 4*x*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] = 4 + 4*x; for(n = 2, N, my(i = n); v[i] = (3 + 4*x)*v[i - 1] + 4*x*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-9a50147d814dfb7b

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-9a50147d814dfb7b

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-09aef78f63ec9a62 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …
  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.