A292219 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - x + n x\right)\,P_{n}(x) = \left(4 - 2\,x - 8\,n - 6\,n x + n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	6	2														8
3	60	40	3													103
4	840	840	126	4												1810
5	15120	20160	4536	288	5											40109
6	332640	554400	166320	15840	550	6										1069756
7	8648640	17297280	6486480	823680	42900	936	7									33299923
8	259459200	605404800	272432160	43243200	3003000	98280	1470	8								1183642118
9	8821612800	23524300800	12350257920	2352430080	204204000	8910720	199920	2176	9							47261918425
10	335221286400	1005663859200	603398315520	134088514560	13967553600	761866560	22790880	372096	3078	10						2093124561904
11	14079294028800	46930980096000	31678411564800	8045310873600	977728752000	63996791040	2393042400	52093440	646380	4200	11					101778167892671
12	647647525324800	2374707592857600	1781030694643200	508865912755200	70675821216000	5397062711040	242175890880	6589820160	109022760	1062600	5566	12				5388573485309818
13	32382376266240000	129529505064960000	106861841678592000	33924394183680000	5300686591200000	462605375232000	24217589088000	790778419200	16353414000	212520000	1669800	7200	13			308486434095022213
14	1748648318376960000	7577476046300160000	6819728441670144000	2381492471694336000	413453554113600000	40593621676608000	2428678219968000	92521075046400	2296019325600	37297260000	390733200	2527200	9126	14		18983915986836677540
15	101421602465863680000	473300811507363840000	461468291219679744000	175797444274163712000	33572428594024320000	3662446755711744000	246510839326752000	10732444705382400	310727948731200	6057075024000	79318839600	684028800	3705156	11368	15	1249480584965869514539

Technical details

Node id: n-c05cf5b45147fec7 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-692d0ae5c5a32a15

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            1;
   2:            6,             2;
   3:           60,            40,            3;
   4:          840,           840,          126,            4;
   5:        15120,         20160,         4536,          288,           5;
   6:       332640,        554400,       166320,        15840,         550,         6;
   7:      8648640,      17297280,      6486480,       823680,       42900,       936,        7;
   8:    259459200,     605404800,    272432160,     43243200,     3003000,     98280,     1470,      8;
   9:   8821612800,   23524300800,  12350257920,   2352430080,   204204000,   8910720,   199920,   2176,    9;
  10: 335221286400, 1005663859200, 603398315520, 134088514560, 13967553600, 761866560, 22790880, 372096, 3078, 10;

%F Formula

The row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(8*n-4-(4*n^2-6*n-2)*x-n*x^2)*P(n-1,x) - (8*x-4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 4 - (4*n^2 - 6*n - 2)*x - n*x^2)*P[n - 1, x] - (8*x - 4*n*x^2)*D[P[n - 1, x], x])/(-2 + (n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n - 4 - (4*n^2 - 6*n - 2)*x - n*x^2)*v[i - 1] - (8*x - 4*n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 1)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - 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

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.