A272867 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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	24	12																											42
2	36	288	576	384	80																									1364
3	216	2592	9072	13824	10080	3456	448																							39688
4	1296	20736	103680	248832	328320	248832	107520	24576	2304																					1086096
5	7776	155520	1010880	3317760	6393600	7713792	5967360	2949120	898560	153600	11264																			28579232
6	46656	1119744	8957952	37324800	94867200	158257152	179601408	140673024	75893760	27648000	6488064	884736	53248																	731815744
7	279936	7838208	74462976	376233984	1188794880	2539579392	3824464896	4151181312	3278112768	1881169920	774918144	222953472	42491904	4816896	245760															18367544448
8	1679616	53747712	591224832	3511517184	13272698880	34613526528	65158152192	90901315584	95378300928	75751096320	45507280896	20511719424	6816595968	1618477056	259522560	25165824	1114112													453973135616
9	10077696	362797056	4534963200	30958682112	136250449920	419877126144	949865398272	1623610884096	2134536634368	2180723834880	1739252072448	1082407256064	522677256192	193523613696	53827338240	10871635968	1504051200	127401984	4980736											11084926454272
10	60466176	2418647040	33861058560	261213880320	1312116019200	4667021328384	12340474675200	25007735439360	39610988789760	49662885888000	49656579489792	39730308710400	25429276753920	12966973931520	5223481344000	1638899122176	391654932480	68797071360	8367636480	629145600	22020096									268013766349824
11	362797056	15963070464	247427592192	2128409395200	12016644710400	48463881928704	146728996356096	344177988599808	638735659499520	951162509721600	1147143293632512	1126813472391168	903809867710464	591834450493440	315425017036800	135971551051776	46925675495424	12764149972992	2672563322880	415236096000	45053116416	3045064704	96468992							6427501315524608
12	2176782336	104485552128	1776254386176	16857002409984	105675526471680	476593431773184	1626161535516672	4335825347149824	9233423070068736	15944340461322240	22560912558784512	26343609452199936	25494293553610752	20489474018377728	13673280338657280	7558798440923136	3445118072782848	1284688991748096	387499985731584	92979666616320	17317836619776	2411691245568	236156092416	14495514624	419430400					153091394969767936
13	13060694016	679156088832	12564387643392	130397969055744	898749890887680	4482430186291200	17021060206166016	50858427634679808	122297297075306496	240457477104599040	390971948895830016	529937346242543616	602084270841790464	575338404913348608	463141746801377280	314036945921507328	179030926093713408	85495991859412992	33999431468580864	11162343513784320	2988608650739712	641290626662400	107541317025792	13567801688064	1210476134400	68048388096	1811939328			3625110741955878912
14	78364164096	4388393189376	87767863787520	988851265339392	7432231465451520	40618967360864256	169913383234043904	562488542962384896	1507909436580298752	3327667628471746560	6117768287968296960	9451998207648202752	12349922157347733504	13705852724875100160	12954909024788152320	10442554457047695360	7177732535894409216	4200888092288090112	2087319370979082240	876422749885562880	308890456144478208	90533135401353216	21783482493566976	4226370225831936	644431321497600	74308839800832	6088116142080	315680096256	7784628224	85408637480690991104

Technical details

Node id: n-a9f53fa0d0ff53f6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4b61633885bd0ded

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:        6,        24,         12;
  2:       36,       288,        576,         384,           80;
  3:      216,      2592,       9072,       13824,        10080,         3456,          448;
  4:     1296,     20736,     103680,      248832,       328320,       248832,       107520,         24576,          2304;
  5:     7776,    155520,    1010880,     3317760,      6393600,      7713792,      5967360,       2949120,        898560,        153600,         11264;
  6:    46656,   1119744,    8957952,    37324800,     94867200,    158257152,    179601408,     140673024,      75893760,      27648000,       6488064,        884736,        53248;
  7:   279936,   7838208,   74462976,   376233984,   1188794880,   2539579392,   3824464896,    4151181312,    3278112768,    1881169920,     774918144,     222953472,     42491904,      4816896,      245760;
  8:  1679616,  53747712,  591224832,  3511517184,  13272698880,  34613526528,  65158152192,   90901315584,   95378300928,   75751096320,   45507280896,   20511719424,   6816595968,   1618477056,   259522560,    25165824,    1114112;
  9: 10077696, 362797056, 4534963200, 30958682112, 136250449920, 419877126144, 949865398272, 1623610884096, 2134536634368, 2180723834880, 1739252072448, 1082407256064, 522677256192, 193523613696, 53827338240, 10871635968, 1504051200, 127401984, 4980736;

%F Formula

The row polynomials satisfy (3+6*n*x+(4*n-2)*x^2)*P(n,x) = (18+(36*n+72)*x+(96*n+96)*x^2+(72*n+48)*x^3+(16*n+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[((18 + (36*n + 72)*x + (96*n + 96)*x^2 + (72*n + 48)*x^3 + (16*n + 8)*x^4)*P[n - 1, x])/(3 + 6*n*x + (4*n - 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] = ((18 + (36*n + 72)*x + (96*n + 96)*x^2 + (72*n + 48)*x^3 + (16*n + 8)*x^4)*v[i - 1])/(3 + 6*n*x + (4*n - 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.