A193722 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4 - 3\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-3\,x + 3\,n x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	12	23	6													41
3	48	197	129	18												392
4	192	1523	1881	621	54											4271
5	768	11237	23379	13716	2727	162										51989
6	3072	80963	267501	248445	84780	11259	486									696506
7	12288	575957	2917899	4032288	2101815	468261	44469	1458								10154435
8	49152	4068563	30906861	61173441	45725904	15202404	2383101	169857	4374							159683657
9	196608	28627397	321274299	887975316	915134787	426023388	98035434	11395728	632043	13122						2689308122
10	786432	200981603	3298625181	12507502005	17306082540	10839848733	3434849712	578999502	51884388	2302911	39366					48221902373
11	3145728	1409230517	33589196619	172493401608	314419826655	257875373547	108086239863	24779536686	3189761370	227043405	8247177	118098				916081121273
12	12582912	9874050803	340119657741	2343181810761	5548197431304	5842891577358	3151523397627	943747136739	163651928418	16607629665	961534233	29111157	354294			18360798203012
13	50331648	69156104357	3430818729819	31481722513116	95800704333147	127659532263714	86862189479868	33048248611356	7413495405921	1005792304869	82515052917	3961715508	101505231	1062882		386858289414353
14	201326592	484243725443	34515655611261	419554848859965	1627256436869700	2712933226010007	2293946765348238	1086792783723504	306722617199856	53420047668702	5822888713785	394128632547	15945355764	350219619	3188646	8541860142453629

Technical details

Node id: n-25f220f6ac2768f4 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-a94fb267242eb635

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      3,        2;
  2:     12,       23,         6;
  3:     48,      197,       129,        18;
  4:    192,     1523,      1881,       621,        54;
  5:    768,    11237,     23379,     13716,      2727,       162;
  6:   3072,    80963,    267501,    248445,     84780,     11259,      486;
  7:  12288,   575957,   2917899,   4032288,   2101815,    468261,    44469,     1458;
  8:  49152,  4068563,  30906861,  61173441,  45725904,  15202404,  2383101,   169857,   4374;
  9: 196608, 28627397, 321274299, 887975316, 915134787, 426023388, 98035434, 11395728, 632043, 13122;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(3*n-3)*x)*P(n-1,x) + (3*n-3)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (3*n - 3)*x)*P[n - 1, x] + (3*n - 3)*x*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (3*n - 3)*x)*v[i - 1] + (3*n - 3)*x*deriv(v[i - 1], x))/(n - 1)); 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.?
  - 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
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.