A272867 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 3\,x^{2} + 6\,n + 6\,n x\right)\,P_{n}(x) = \left(12 + 72\,x + 144\,x^{2} + 108\,x^{3} + 27\,x^{4} + 36\,n + 144\,n x + 162\,n x^{2} + 54\,n x^{3}\right)\,P_{n-1}(x)\)

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

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	24	36	9																											69
2	252	1080	1404	648	81																									3465
3	2160	15552	41796	52488	32076	8748	729																							153549
4	16848	171072	707616	1539648	1907064	1364688	542376	104976	6561																					6360849
5	124416	1632960	9156240	28693440	55287360	68024448	53800200	26768880	7873200	1180980	59049																			252601173
6	886464	14276736	101196864	415704960	1097524080	1953813312	2395342368	2029396032	1172713140	446410440	104162436	12754584	531441																	9744712857
7	6158592	117573120	1008189504	5132066688	17274430656	40553908416	68284263600	83545359552	74421580464	47855199168	21740188428	6696156600	1294590276	133923132	4782969															368068371165
8	41990400	927148032	9331946496	56717273088	232477330176	679972382208	1465019861760	2366978692608	2892518571744	2676830053248	1864924254144	964246550400	360521071344	93210499872	15382028304	1377495072	43046721													13680520195617
9	282175488	7074542592	81765386496	577935710208	2794988519424	9805316032512	25818951336192	52095026962944	81529341661440	99559833823872	94941092847456	70401018139776	40189796220672	17356437907200	5510669035536	1227348109152	176663742984	13947137604	387420489											502087876712037
10	1874451456	52605573120	686593428480	5534469089280	30862011813120	126391601166336	393951346191360	955797913958400	1831085832608640	2794055904241920	3410506988883072	3331940179956480	2597163646125600	1603486913512320	774469054330560	286976303339904	79150005902700	15481322740440	1975844493900	139471376040	3486784401									18239713369967529
11	12335099904	383113691136	5573106975744	50463256504320	318852355599360	1493909935524864	5384290290990336	15286272193234944	34728193559600640	63793008979983360	95358038565829248	116367775233322752	115975668506087808	94170329254869120	61958578130174880	32730919374865536	13690454483535984	4440545458911936	1082367613758420	188704771782120	21632010423804	1380766622796	31381059609							657047384678448621
12	80540946432	2742745743360	44001478139904	442208977993728	3122939280992256	16480042482511872	67467578100848640	219609124426543104	577952224231143168	1244000768296974336	2207040280375163904	3243233905303910400	3957264569913308928	4010354144644916736	3369188385040718592	2336654786558197248	1328546029121855280	612861550127741376	225956014242992352	65150092330006464	14224657748044392	2236841928929520	232721938060344	13556617751088	282429536481					23502079528882969905
13	522427760640	19355948531712	339493149904896	3749960344485888	29275362787147776	171856373342404608	788040642698090496	2894705788486127616	8667854548438745088	21420229052873604096	44068012409381545728	75928991907564177408	109983165711756060672	134188991174095497216	137933823060348486912	119272159302184751616	86473459070105792256	52282650849239663424	26152454403599567472	10701913386371448384	3526522139152913472	915017471727435648	180906285579393816	25906696522329168	2467304430698016	132177023073108	2541865828329			835612646991445465461
14	3369659056128	134943090573312	2568855663230976	30934880690208768	264567037189238784	1710339492766334976	8685802097031241728	35553392363285102592	119428058001215867904	333502987980385327104	781514569634050796544	1547230159794521180160	2600154092306714082048	3720319657575130294272	4539062922467056247808	4723092461979458316288	4186360412495714456640	3152415655796518290816	2008102340723591496384	1075372991250109808256	479954450468557144080	176420021180261766336	52541247851714250720	12390802850965436352	2238153531696937764	294226053360738408	25835524279135956	1281100377477816	22876792454961	29556670094481846493881

Technical details

Node id: n-eb325714e1fa06da · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8b65e409dcd1ef0f

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:         1;
  1:        24,         36,           9;
  2:       252,       1080,        1404,          648,            81;
  3:      2160,      15552,       41796,        52488,         32076,          8748,            729;
  4:     16848,     171072,      707616,      1539648,       1907064,       1364688,         542376,         104976,           6561;
  5:    124416,    1632960,     9156240,     28693440,      55287360,      68024448,       53800200,       26768880,        7873200,        1180980,          59049;
  6:    886464,   14276736,   101196864,    415704960,    1097524080,    1953813312,     2395342368,     2029396032,     1172713140,      446410440,      104162436,       12754584,         531441;
  7:   6158592,  117573120,  1008189504,   5132066688,   17274430656,   40553908416,    68284263600,    83545359552,    74421580464,    47855199168,    21740188428,     6696156600,     1294590276,      133923132,       4782969;
  8:  41990400,  927148032,  9331946496,  56717273088,  232477330176,  679972382208,  1465019861760,  2366978692608,  2892518571744,  2676830053248,  1864924254144,   964246550400,   360521071344,    93210499872,   15382028304,    1377495072,     43046721;
  9: 282175488, 7074542592, 81765386496, 577935710208, 2794988519424, 9805316032512, 25818951336192, 52095026962944, 81529341661440, 99559833823872, 94941092847456, 70401018139776, 40189796220672, 17356437907200, 5510669035536, 1227348109152, 176663742984, 13947137604, 387420489;

%F Formula

The row polynomials satisfy (6*n-4+6*n*x+3*x^2)*P(n,x) = (36*n+12+(144*n+72)*x+(162*n+144)*x^2+(54*n+108)*x^3+27*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[((36*n + 12 + (144*n + 72)*x + (162*n + 144)*x^2 + (54*n + 108)*x^3 + 27*x^4)*P[n - 1, x])/(6*n - 4 + 6*n*x + 3*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] = ((36*n + 12 + (144*n + 72)*x + (162*n + 144)*x^2 + (54*n + 108)*x^3 + 27*x^4)*v[i - 1])/(6*n - 4 + 6*n*x + 3*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 (−∞, 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_A013610 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.