A200536 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 38\,x + 86\,x^{2} + 82\,x^{3} + 28\,x^{4}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	6	26	28																											60
2	36	312	1012	1456	784																									3600
3	216	2808	15192	43784	70896	61152	21952																							216000
4	1296	22464	170208	736320	1989136	3436160	3706752	2283008	614656																					12960000
5	7776	168480	1641600	9472320	35844960	92951456	167276480	206286080	166835200	79905280	17210368																			777600000
6	46656	1213056	14447808	104232960	507314880	1754902656	4424055616	8189545728	11048190720	10593157120	6852185088	2684817408	481890304																	46656000000
7	279936	8491392	119532672	1035006336	6158484864	26638125696	86376619392	213299994752	403090890496	580119181824	625884535808	490874116096	264557776896	87704035328	13492928512															2799360000000
8	1679616	58226688	945810432	9555646464	67207988736	348929538048	1383288560640	4271459592192	10382890549504	19933478096896	30124950876160	35461579718656	31874840682496	21149201661952	9768880242688	2806529130496	377801998336													167961600000000
9	10077696	393030144	7235785728	83555297280	678177432576	4108543036416	19253723037696	71364287195136	212087372391936	509156891449856	989741071162368	1554155587805184	1956748741312512	1948565299838976	1500988063776768	863007707627520	348765216178176	88405667610624	10578455953408											10077696000000000
10	60466176	2620200960	53915673600	700467056640	6444104325120	44623419789312	241333425285120	1043821727170560	3667099946480640	10567413072353280	25114972031644672	49314594337648640	79861287723356160	106083215531704320	114457590935224320	98763564299452416	66558457481134080	33762545439866880	12127444146585600	2750398547886080	296196766695424									604661760000000000
11	362797056	17293326336	393312319488	5677975480320	58386408284160	454900308774912	2788644387336192	13787055174537216	55899300493301760	187976085403392000	528121370572511232	1244764404874545152	2464566396005052416	4093701415451648000	5681025205689630720	6538808785000267776	6172015962858848256	4698474967533355008	2814227655787806720	1277167211414814720	412855978949607424	84712275274891264	8293509467471872							36279705600000000000
12	2176782336	113192681472	2819658719232	44778186326016	508958557138944	4406431781486592	30194093784121344	167964293763661824	771941280343191552	2967275870133239808	9621286857735708672	26463072455427538944	61938671278798802944	123494338125328515072	209530247124022099968	301563147690578149376	366109829946469711872	371749910819537682432	311862162050641821696	212390221409905803264	114481857732541415424	47003211023953952768	13812247624541011968	2587574953851224064	232218265089212416					2176782336000000000000
13	13060694016	735752429568	19921911939072	345149639737344	4296934631448576	40925302391660544	309982628623269888	1916212290850750464	9844153945869754368	42577128735104950272	156491249619487850496	492015617397424373760	1329067943530508673024	3092337510752711057408	6202317069809040474112	10715006778877241917440	15904147739106468954112	20193123178861626785792	21787745892112982605824	19791755044723177291776	14941177440470770384896	9205473766667163009024	4510448988881208475648	1690733796631884333056	455413191877815435264	78489773600153796608	6502111422497947648			130606940160000000000000
14	78364164096	4754092621824	139026734235648	2609468616867840	35313311956156416	366936304680271872	3044267803603353600	20702730556276015104	117565956838789595136	565064719147064328192	2321589155314008932352	8213025799074168963072	25148568702861745569792	66886228883437374308352	154828580117278973181952	312135734789374413438976	547679940640100236853248	834686770098500727209984	1101063814698061322780672	1250639372471402161504256	1214289580784790744858624	997872597304491972231168	684757980199815059865600	385169341957383290617856	172984129552369722589184	59652227936116885422080	14831316154717818585088	2366768557789252943872	182059119829942534144	7836416409600000000000000

Technical details

Node id: n-61ae048ec6ed2266 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-be5ce6bf66b9c063

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        6,        26,         28;
  2:       36,       312,       1012,        1456,          784;
  3:      216,      2808,      15192,       43784,        70896,         61152,          21952;
  4:     1296,     22464,     170208,      736320,      1989136,       3436160,        3706752,        2283008,          614656;
  5:     7776,    168480,    1641600,     9472320,     35844960,      92951456,      167276480,      206286080,       166835200,        79905280,        17210368;
  6:    46656,   1213056,   14447808,   104232960,    507314880,    1754902656,     4424055616,     8189545728,     11048190720,     10593157120,      6852185088,       2684817408,        481890304;
  7:   279936,   8491392,  119532672,  1035006336,   6158484864,   26638125696,    86376619392,   213299994752,    403090890496,    580119181824,    625884535808,     490874116096,     264557776896,      87704035328,      13492928512;
  8:  1679616,  58226688,  945810432,  9555646464,  67207988736,  348929538048,  1383288560640,  4271459592192,  10382890549504,  19933478096896,  30124950876160,   35461579718656,   31874840682496,   21149201661952,    9768880242688,   2806529130496,    377801998336;
  9: 10077696, 393030144, 7235785728, 83555297280, 678177432576, 4108543036416, 19253723037696, 71364287195136, 212087372391936, 509156891449856, 989741071162368, 1554155587805184, 1956748741312512, 1948565299838976, 1500988063776768, 863007707627520, 348765216178176, 88405667610624, 10578455953408;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+38*x+86*x^2+82*x^3+28*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 38*x + 86*x^2 + 82*x^3 + 28*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 38*x + 86*x^2 + 82*x^3 + 28*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.