A272867 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,x + 2\,x^{2} + 6\,n + 4\,n x\right)\,P_{n}(x) = \left(18 + 72\,x + 96\,x^{2} + 48\,x^{3} + 8\,x^{4} + 36\,n + 96\,n x + 72\,n x^{2} + 16\,n x^{3}\right)\,P_{n-1}(x)\)

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

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	18	20	4																											42
2	180	528	480	160	16																									1364
3	1512	7344	13392	11520	4896	960	64																							39688
4	11664	79488	221184	324864	273024	133632	36864	5120	256																					1086096
5	85536	751680	2825280	5944320	7706880	6400512	3432960	1167360	238080	25600	1024																			28579232
6	606528	6531840	30979584	85224960	150854400	180320256	148746240	85045248	33315840	8663040	1400832	122880	4096																	731815744
7	4199040	53561088	306996480	1045094400	2354075136	3701376000	4178635776	3432775680	2057287680	892588032	274821120	57802752	7741440	573440	16384															18367544448
8	28553472	421023744	2830712832	11496038400	31499145216	61618765824	88846958592	96143376384	78773575680	48945364992	22939435008	8001552384	2027225088	357826560	40894464	2621440	65536													453973135616
9	191476224	3204707328	24730665984	116740030464	377147695104	884149862400	1556354580480	2100317257728	2199822188544	1799210631168	1150320476160	572671918080	219906441216	64061964288	13780647936	2095054848	208797696	11796480	262144											11084926454272
10	1269789696	23783362560	207197429760	1114996285440	4151607644160	11355386609664	23644693463040	38335734743040	49097818275840	50108633579520	40933684150272	26784037601280	13996859719680	5801461678080	1884700016640	471047602176	88013537280	11749294080	1038090240	52428800	1048576									268013766349824
11	8344332288	172933263360	1678782910464	10145418117120	42789897216000	133845024817152	322117025218560	610787279241216	927016887582720	1137973790638080	1137198103658496	928187154432000	619084397150208	336670571888640	148501663580160	52663563583488	14815364382720	3241021538304	535135518720	63669534720	5052039168	230686720	4194304							6427501315524608
12	54419558400	1236412366848	13234836602880	88754671779840	418296298438656	1473284983357440	4026155969544192	8749465341788160	15375842311864320	22104354554118144	26199300715315200	25728494837170176	20987039664046080	14225262174535680	7997925560942592	3714888971059200	1415971385376768	438476860293120	108725419376640	21142161063936	3126727802880	335510765568	24159191040	1006632960	16777216					153091394969767936
13	352638738432	8715836473344	101986606006272	751599404974080	3915108466753536	15336551874428928	46932704209797120	115066011965718528	230000317958062080	379474792161214464	521318805167407104	599931541681864704	580556979242532864	473361230698905600	325285753254838272	188127004267118592	91277445480579072	36959765032599552	12393670099599360	3404411128774656	754455750377472	132015181529088	17678186053632	1727382159360	113850187776	4362076160	67108864			3625110741955878912
14	2272560758784	60706105786368	770894403600384	6192997988696064	35335585760477184	152409870665515008	516450815554093056	1410675011497230336	3162446358577938432	5894374297363808256	9220492443795849216	12187594662548078592	13676857256630550528	13070237903483830272	10653555405134757888	7408172628524924928	4389724981429272576	2210843128510808064	942460526653341696	338002524298543104	101127062223323136	24951517077307392	4997379400925184	794752828047360	97228060360704	8720662659072	529891590144	18790481920	268435456	85408637480690991104

Technical details

Node id: n-e5977c2b696b104a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a6f3cf41f153a3d4

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:        18,         20,           4;
  2:       180,        528,         480,          160,           16;
  3:      1512,       7344,       13392,        11520,         4896,          960,            64;
  4:     11664,      79488,      221184,       324864,       273024,       133632,         36864,          5120,           256;
  5:     85536,     751680,     2825280,      5944320,      7706880,      6400512,       3432960,       1167360,        238080,         25600,          1024;
  6:    606528,    6531840,    30979584,     85224960,    150854400,    180320256,     148746240,      85045248,      33315840,       8663040,       1400832,       122880,         4096;
  7:   4199040,   53561088,   306996480,   1045094400,   2354075136,   3701376000,    4178635776,    3432775680,    2057287680,     892588032,     274821120,     57802752,      7741440,      573440,       16384;
  8:  28553472,  421023744,  2830712832,  11496038400,  31499145216,  61618765824,   88846958592,   96143376384,   78773575680,   48945364992,   22939435008,   8001552384,   2027225088,   357826560,    40894464,    2621440,     65536;
  9: 191476224, 3204707328, 24730665984, 116740030464, 377147695104, 884149862400, 1556354580480, 2100317257728, 2199822188544, 1799210631168, 1150320476160, 572671918080, 219906441216, 64061964288, 13780647936, 2095054848, 208797696, 11796480, 262144;

%F Formula

The row polynomials satisfy (6*n-3+(4*n+2)*x+2*x^2)*P(n,x) = (36*n+18+(96*n+72)*x+(72*n+96)*x^2+(16*n+48)*x^3+8*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 + 18 + (96*n + 72)*x + (72*n + 96)*x^2 + (16*n + 48)*x^3 + 8*x^4)*P[n - 1, x])/(6*n - 3 + (4*n + 2)*x + 2*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 + 18 + (96*n + 72)*x + (72*n + 96)*x^2 + (16*n + 48)*x^3 + 8*x^4)*v[i - 1])/(6*n - 3 + (4*n + 2)*x + 2*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_A013609 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.