A272867 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 24\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 24 + 32\,x\)

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	P_n(1)
0																													0
1	24	32																											56
2	288	1536	2304	1024																									5152
3	2592	24192	82944	129024	92160	24576																							355488
4	20736	276480	1492992	4202496	6635520	5898240	2752512	524288																					21803264
5	155520	2695680	19906560	81838080	205701120	327352320	330301440	204472320	70778880	10485760																			1253687680
6	1119744	23887872	223948800	1214300160	4220190720	9852420096	15755378688	17270046720	12740198400	6039797760	1660944384	201326592																	69203559936
7	7838208	198567936	2257403904	15216574464	67722117120	209799217152	464932306944	745952772096	866843099136	721378344960	418557984768	160658620416	36641439744	3758096384															3713924383232
8	53747712	1576599552	21069103104	169890545664	923027374080	3574390063104	10180947345408	21703862255616	34906105184256	42363141488640	38507334598656	25773025001472	12311523753984	3968549781504	773094113280	68719476736													195246310432768
9	362797056	12093235200	185752092672	1744005758976	11196723363840	52106901848064	181844419018752	485725669687296	1004877543112704	1619085565624320	2032039222050816	1976202499719168	1472106483154944	823113302409216	333976656936960	92771293593600	15771119910912	1236950581248											10103996564895744
10	2418647040	90296156160	1567283281920	16795085045760	124453902090240	676963182182400	2800866369208320	9013700560158720	22884657817190400	46225761270497280	74587032885657600	96146139308359680	98637918272225280	79876083784089600	50346981033246720	24157644851773440	8516404751892480	2078076976496640	313360813916160	21990232555520									516426491094671360
11	15963070464	659806912512	12770456371200	153813052293120	1292370184765440	8049133514391552	38547934723178496	145347847850557440	438295684479713280	1067886120617902080	2115404492169019392	3417235585977286656	4502000117124956160	4823395247188869120	4177046048310558720	2894419829899395072	1580076961634451456	663722942886051840	206818137184665600	44992015808593920	6095692464390144	387028092977152							26131180449390370816
12	104485552128	4736678363136	101142014459904	1352646738837504	12709158180618240	89206575662628864	485612438880780288	2101116716388974592	7347152084577288192	21002158600177582080	49455736144930013184	96391962826698129408	155860501790082465792	209088156730656817152	232206288105158737920	212498125193314566144	159031935774939414528	96234438561745600512	46310717278390321152	17294350334684037120	4827788431796994048	947444771608068096	116530640358211584	6755399441055744					1311310146187589517312
13	679156088832	33505033715712	782387814334464	11503998603362304	119531471634432000	933726731309678592	5696143895084138496	27829429378914189312	110802805449799237632	363959341517572669440	994869044679335215104	2276434313878129606656	4376517054975140954112	7082240087620847665152	9647214978708705116160	11042796021155390029824	10583567831386036371456	8443655017483890327552	5559668618347205885952	2984555528150518333440	1283752833000564326400	431449629376697597952	109072679375286042624	19496082786886877184	2195504818343116800	117093590311632896			65346955618348210946048
14	4388393189376	234047636766720	5933107592036352	95132562757779456	1083172462956380160	9320962737410408448	62998716811787108352	343133169568494649344	1533389243199780814848	5695086115345032806400	17744551301824892633088	46694105682919676706816	104258463699164190474240	198103014164653815103488	320795272920505201459200	442729298326942119886848	520028870475029850095616	518379393965627875000320	436523034157401891667968	308471541849668934696960	181231666610799667838976	87394089158580639891456	33976139625401602277376	10379292938813451534336	2397491261630683545600	393434463447086530560	40856655819505139712	2017612633061982208	3237187647555095988404224

Technical details

Node id: n-ba35e8a0ffd8c223 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-70c371a2e54a122c

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:         0;
  1:        24,          32;
  2:       288,        1536,         2304,          1024;
  3:      2592,       24192,        82944,        129024,          92160,          24576;
  4:     20736,      276480,      1492992,       4202496,        6635520,        5898240,         2752512,          524288;
  5:    155520,     2695680,     19906560,      81838080,      205701120,      327352320,       330301440,       204472320,         70778880,         10485760;
  6:   1119744,    23887872,    223948800,    1214300160,     4220190720,     9852420096,     15755378688,     17270046720,      12740198400,       6039797760,       1660944384,        201326592;
  7:   7838208,   198567936,   2257403904,   15216574464,    67722117120,   209799217152,    464932306944,    745952772096,     866843099136,     721378344960,     418557984768,     160658620416,      36641439744,      3758096384;
  8:  53747712,  1576599552,  21069103104,  169890545664,   923027374080,  3574390063104,  10180947345408,  21703862255616,   34906105184256,   42363141488640,   38507334598656,   25773025001472,   12311523753984,   3968549781504,    773094113280,    68719476736;
  9: 362797056, 12093235200, 185752092672, 1744005758976, 11196723363840, 52106901848064, 181844419018752, 485725669687296, 1004877543112704, 1619085565624320, 2032039222050816, 1976202499719168, 1472106483154944, 823113302409216, 333976656936960, 92771293593600, 15771119910912, 1236950581248;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+24*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 24 + 32*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 24 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 24*n*x + 16*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 24 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 24*n*x + 16*n*x^2)*v[i - 1])/(n - 1)); 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

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.