A038333 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 42\,x + 21\,n x\right)\,P_{n}(x) = \left(361 + 1596\,x + 1323\,x^{2} + 399\,n x + 441\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 361 + 1596\,x + 1323\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	361	1596	1323															3280
1	6859	45486	75411	37044														164800
2	130321	1152312	2865618	2815344	972405													7936000
3	2476099	27367410	90744570	133728840	92378475	24504606												371200000
4	47045881	623976948	2586220245	5081695920	5265573075	2793525084	600362847											16998400000
5	893871739	13831489014	68793458517	168966389340	233440406325	185769418086	79848258651	14408708328										765952000000
6	16983563041	300340904304	1742767615764	5136578235936	8870735440350	9412317183024	6068467657476	2190123665856	340405734249									34078720000000
7	322687697779	6419786829498	42573323185092	146392479724176	303379152059970	402376559574276	345902656476132	187255573430688	58209380556579	7942800465810								1500774400000000
8	6131066257801	135528833067180	1011116425645935	3973510163941920	9607006481899050	15290309263822488	16430376182616270	11859519650610240	5529891152875005	1509132088503900	183478690760211							65536000000000000
9	116490258898219	2832552611104062	23480370328888935	103807953032982660	286837764959557350	532612439356483332	686789724433360086	619659901744385040	385249083650292015	157704303248657550	38347046368884099	4203330006506652						2841640960000000000
10	2213314919066161	58711090484702376	535352443498667718	2629801476835560720	8174876301347384475	17347948024754028528	26098009528467683268	28256491519543957824	21959197768066644855	11985527046897973800	4371563286052787286	958359241483516656	95625757648026333					122473676800000000000
11	42052983462257059	1208469945810123906	12021095776742811486	64956096477838349784	224354938492533773925	535617895264280630802	920886907647359681028	1163225567554559597088	1084784369742492255837	740106295145949882150	359925377218346153214	118357366323214307016	23619562139062504251	2162613288347672454				5251268608000000000000
12	799006685782884121	24727154275807150692	266467623051132321273	1570756514827727367504	5967841363901398386405	15830484460033183088148	30619489679274709394181	44202571567073264689344	48092107058583823342107	39373654901764533730380	23935037585020019188731	10494353147325001888752	3141401764495313065383	575255134700480872764	48658798987822630215			224143605760000000000000
13	15181127029874798299	503374212043216996230	5841790197659439350985	37305467227158524978220	154621344428354412738675	451168807110945718012218	969617173177032464149065	1574716612076985054557880	1958035787385198521785785	1870248607833815352193050	1364297142346141093757667	747722661746906384573580	298433167627054741211385	81973856694818524368870	13867757711529449611275	1089957097327226916816		9529458688000000000000000
14	288441413567621167681	10201717364075864456928	126850301434890683049960	872374002850476276413760	3917074058851645122713100	12468665214702499843246752	29476362064581786910131576	53190427785711495176177280	74405359920637543827859830	81222225254497123866669600	69124388545537815417055128	45461537834211908182073664	22680920739656160332065260	8306684145074943802712160	2107899172152476340913800	331346957587476982712064	24319667734113750581457	403726925824000000000000000

Technical details

Node id: n-7b81009382daeb8b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e10af1e97a00021b

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:             361,             1596,              1323;
  1:            6859,            45486,             75411,              37044;
  2:          130321,          1152312,           2865618,            2815344,             972405;
  3:         2476099,         27367410,          90744570,          133728840,           92378475,           24504606;
  4:        47045881,        623976948,        2586220245,         5081695920,         5265573075,         2793525084,          600362847;
  5:       893871739,      13831489014,       68793458517,       168966389340,       233440406325,       185769418086,        79848258651,        14408708328;
  6:     16983563041,     300340904304,     1742767615764,      5136578235936,      8870735440350,      9412317183024,      6068467657476,      2190123665856,       340405734249;
  7:    322687697779,    6419786829498,    42573323185092,    146392479724176,    303379152059970,    402376559574276,    345902656476132,    187255573430688,     58209380556579,      7942800465810;
  8:   6131066257801,  135528833067180,  1011116425645935,   3973510163941920,   9607006481899050,  15290309263822488,  16430376182616270,  11859519650610240,   5529891152875005,   1509132088503900,   183478690760211;
  9: 116490258898219, 2832552611104062, 23480370328888935, 103807953032982660, 286837764959557350, 532612439356483332, 686789724433360086, 619659901744385040, 385249083650292015, 157704303248657550, 38347046368884099, 4203330006506652;

%F Formula

The row polynomials satisfy (19+(21*n+42)*x)*P(n,x) = (361+(399*n+1596)*x+(441*n+1323)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1596*x + 1323*x^2.

%t Mathematica

P[0, x_] := 361 + 1596*x + 1323*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((361 + (399*n + 1596)*x + (441*n + 1323)*x^2)*P[n - 1, x])/(19 + (21*n + 42)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 1596*x + 1323*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (399*n + 1596)*x + (441*n + 1323)*x^2)*v[i - 1])/(19 + (21*n + 42)*x)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.