A038328 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 8\,x + 4\,n x\right)\,P_{n}(x) = \left(98 + 224\,x + 96\,x^{2} + 56\,n x + 32\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 448\,x + 192\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	196	448	192								836
1	2744	9408	8064	2048							22264
2	38416	175616	225792	114688	20480						574992
3	537824	3073280	5268480	4014080	1433600	196608					14523872
4	7529536	51631104	110638080	112394240	60211200	16515072	1835008				360754240
5	105413504	843308032	2168506368	2753658880	1966899200	809238528	179830784	16777216			8843632512
6	1475789056	13492928512	40478785536	61681958912	55073177600	30211571712	10070523904	1879048192	150994944		214514778368
7	20661046784	212513624064	728618139648	1295321137152	1387844075520	951664508928	422962003968	118380036096	19025362944	1342177280	5158332112384

All 15 rows

Technical details

Node id: n-f37732e25746403d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-604d756e1453b62f

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:           196,            448,             192;
  1:          2744,           9408,            8064,            2048;
  2:         38416,         175616,          225792,          114688,           20480;
  3:        537824,        3073280,         5268480,         4014080,         1433600,          196608;
  4:       7529536,       51631104,       110638080,       112394240,        60211200,        16515072,         1835008;
  5:     105413504,      843308032,      2168506368,      2753658880,      1966899200,       809238528,       179830784,        16777216;
  6:    1475789056,    13492928512,     40478785536,     61681958912,     55073177600,     30211571712,     10070523904,      1879048192,      150994944;
  7:   20661046784,   212513624064,    728618139648,   1295321137152,   1387844075520,    951664508928,    422962003968,    118380036096,    19025362944,     1342177280;
  8:  289254654976,  3305767485440,  12750817443840,  25906422743040,  32383028428800,  26646606249984,  14803670138880,   5524401684480,  1331775406080,   187904819200,   11811160064;
  9: 4049565169664, 50908819275776, 218180654039040, 498698637803520, 712426625433600, 683929560416256, 455953040277504, 212689464852480, 68364470845440, 14468671078400, 1818918649856, 103079215104;

%F Formula

The row polynomials satisfy (7+(4*n+8)*x)*P(n,x) = (98+(56*n+224)*x+(32*n+96)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 448*x + 192*x^2.

%t Mathematica

P[0, x_] := 196 + 448*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (56*n + 224)*x + (32*n + 96)*x^2)*P[n - 1, x])/(7 + (4*n + 8)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 448*x + 192*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (56*n + 224)*x + (32*n + 96)*x^2)*v[i - 1])/(7 + (4*n + 8)*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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

Related ?

Same fingerprint ?

A038255 → basis A013611 → θ+1

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.