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	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

All 15 rows

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.