A038333 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(12 + 14\,x + 7\,n x\right)\,P_{n}(x) = \left(144 + 336\,x + 147\,x^{2} + 84\,n x + 49\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 144 + 336\,x + 147\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	144	336	147								627
1	1728	6048	5292	1372							14440
2	20736	96768	127008	65856	12005						322373
3	248832	1451520	2540160	1975680	720300	100842					7037334
4	2985984	20901888	45722880	47416320	25930800	7260624	823543				151042039
5	35831808	292626432	768144384	995742720	726062400	304946208	69177612	6588344			3199119908
6	429981696	4013162496	12290310144	19118260224	17425497600	9758278656	3320525376	632481024	51883209		67040380425
7	5159780352	54177693696	189621927936	344128684032	376390748160	263473523712	119538913536	34153975296	5603386572	403536070	1392652169362

All 15 rows

Technical details

Node id: n-5e5215f6b5f47c3a · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1b5229a076e1d872

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:          144,           336,            147;
  1:         1728,          6048,           5292,           1372;
  2:        20736,         96768,         127008,          65856,           12005;
  3:       248832,       1451520,        2540160,        1975680,          720300,          100842;
  4:      2985984,      20901888,       45722880,       47416320,        25930800,         7260624,         823543;
  5:     35831808,     292626432,      768144384,      995742720,       726062400,       304946208,       69177612,        6588344;
  6:    429981696,    4013162496,    12290310144,    19118260224,     17425497600,      9758278656,     3320525376,      632481024,       51883209;
  7:   5159780352,   54177693696,   189621927936,   344128684032,    376390748160,    263473523712,   119538913536,    34153975296,     5603386572,     403536070;
  8:  61917364224,  722369249280,  2844328919040,  5899348869120,   7527814963200,   6323364569088,  3586167406080,  1366159011840,   336203194320,   48424328400,   3107227739;
  9: 743008370688, 9535274090496, 41716824145920, 97339256340480, 141953082163200, 139114020519936, 94674819520512, 45083247390720, 14792940550080, 3196005674400, 410154061548, 23727920916;

%F Formula

The row polynomials satisfy (12+(7*n+14)*x)*P(n,x) = (144+(84*n+336)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 336*x + 147*x^2.

%t Mathematica

P[0, x_] := 144 + 336*x + 147*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((144 + (84*n + 336)*x + (49*n + 147)*x^2)*P[n - 1, x])/(12 + (7*n + 14)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 336*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144 + (84*n + 336)*x + (49*n + 147)*x^2)*v[i - 1])/(12 + (7*n + 14)*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 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.