A272867 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 8\,x^{2} + 12\,n x + 16\,n x^{2}\right)\,P_{n}(x) = \left(18 + 144\,x + 384\,x^{2} + 384\,x^{3} + 128\,x^{4} + 72\,n x + 384\,n x^{2} + 576\,n x^{3} + 256\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	48	48																											102
2	36	576	2304	3072	1280																									7268
3	216	5184	36288	110592	161280	110592	28672																							452824
4	1296	41472	414720	1990656	5253120	7962624	6881280	3145728	589824																					26280720
5	7776	311040	4043520	26542080	102297600	246841344	381911040	377487360	230031360	78643200	11534336																			1459650656
6	46656	2239488	35831808	298598400	1517875200	5064228864	11494490112	18006147072	19428802560	14155776000	6643777536	1811939328	218103808																	78677856832
7	279936	15676416	297851904	3009871872	19020718080	81266540544	244765753344	531351207936	839196868608	963158999040	793516179456	456608710656	174046838784	39460012032	4026531840															4149742040448
8	1679616	107495424	2364899328	28092137472	212363182080	1107632848896	4170121740288	11635368394752	24416845037568	38784561315840	46599455637504	42008001380352	27920777084928	13258564042752	4252017623040	824633720832	73014444032													215293922664704
9	10077696	725594112	18139852800	247669456896	2180007198720	13436068036608	60791385489408	207822193164288	546441378398208	1116530603458560	1780994122186752	2216770060419072	2140886041362432	1585345443397632	881907109724160	356241767399424	98569499443200	16698832846848	1305670057984											11026186727564800
10	60466176	4837294080	135444234240	2089711042560	20993856307200	149344682508288	789790379212800	3200990136238080	10140413130178560	25427397574656000	50848337397547008	81367672238899200	104158317584056320	106225450447011840	85581518340096000	53703446435463168	25667497655009280	9017369737297920	2193525697413120	329853488332800	23089744183296									558847238577447936
11	362797056	31926140928	989710368768	17027275161600	192266315366400	1550844221718528	9390655766790144	44054782540775424	163516328831877120	486995204977459200	1174674732679692288	2307713991457112064	3702005218142060544	4848307818442260480	5167923479130931200	4455515784864595968	3075321069268107264	1673022665260007424	700596439713054720	217703302299648000	47241616599023616	6385963534123008	404620279021568							28082534833598093312
12	2176782336	208971104256	7105017544704	134856019279872	1690808423546880	15250989816741888	104074338273067008	554985644435177472	2363756305937596416	8163502316196986880	23102374460195340288	53951712158105468928	104424626395589640192	167849771158550347776	224023025068560875520	247686707312169320448	225779258017896726528	168386755526406438912	101580796259620356096	48748123450937180160	18159067851418238976	5057683119025422336	990510443044798464	121597189939003392	7036874417766400					1401072447861144752128
13	13060694016	1358312177664	50257550573568	1043183752445952	14379998254202880	143437765961318400	1089347853194625024	6509878737239015424	31308108051278462976	123114228277554708480	400355275669329936384	1085311685104729325568	2466137173367973740544	4713172213050151796736	7588114379593765355520	10290362643955952123904	11732970772477601906688	11206130644996979687424	8912746962899662012416	5852282756154953564160	3133783304558044250112	1344883920286305484800	451060976166547488768	113814969782907174912	20308419569673830400	2283325011076841472	121597189939003392			69476021495331751747584
14	78364164096	8776786378752	351071455150080	7910810122715136	118915703447224320	1299806955547656192	10874456526978809856	71998533499185266688	386024815764556480512	1703765825777534238720	6264594726879536087040	19357692329263519236096	50585281156496316432384	112278345522176820510720	212253229462129087610880	342181624448538881556480	470399879472376002379776	550618804032384547160064	547178249185940534722560	459497930692001991229440	323895118942152381431808	189861745973218699640832	91366547756697941704704	35453363087375584985088	10811763477930678681600	2493390912095910887424	408566558195051397120	42369865294301626368	2089670227099910144	3427124677896338865274880

Technical details

Node id: n-4670037a8155e6a9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1d2b3853bdcde690

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,        48,          48;
  2:       36,       576,        2304,         3072,          1280;
  3:      216,      5184,       36288,       110592,        161280,         110592,          28672;
  4:     1296,     41472,      414720,      1990656,       5253120,        7962624,        6881280,         3145728,          589824;
  5:     7776,    311040,     4043520,     26542080,     102297600,      246841344,      381911040,       377487360,       230031360,         78643200,         11534336;
  6:    46656,   2239488,    35831808,    298598400,    1517875200,     5064228864,    11494490112,     18006147072,     19428802560,      14155776000,       6643777536,       1811939328,        218103808;
  7:   279936,  15676416,   297851904,   3009871872,   19020718080,    81266540544,   244765753344,    531351207936,    839196868608,     963158999040,     793516179456,     456608710656,     174046838784,      39460012032,      4026531840;
  8:  1679616, 107495424,  2364899328,  28092137472,  212363182080,  1107632848896,  4170121740288,  11635368394752,  24416845037568,   38784561315840,   46599455637504,   42008001380352,   27920777084928,   13258564042752,   4252017623040,    824633720832,    73014444032;
  9: 10077696, 725594112, 18139852800, 247669456896, 2180007198720, 13436068036608, 60791385489408, 207822193164288, 546441378398208, 1116530603458560, 1780994122186752, 2216770060419072, 2140886041362432, 1585345443397632, 881907109724160, 356241767399424, 98569499443200, 16698832846848, 1305670057984;

%F Formula

The row polynomials satisfy (3+12*n*x+(16*n-8)*x^2)*P(n,x) = (18+(72*n+144)*x+(384*n+384)*x^2+(576*n+384)*x^3+(256*n+128)*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 + (72*n + 144)*x + (384*n + 384)*x^2 + (576*n + 384)*x^3 + (256*n + 128)*x^4)*P[n - 1, x])/(3 + 12*n*x + (16*n - 8)*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 + (72*n + 144)*x + (384*n + 384)*x^2 + (576*n + 384)*x^3 + (256*n + 128)*x^4)*v[i - 1])/(3 + 12*n*x + (16*n - 8)*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_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.