A095660 → basis A008277

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 3\)

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	3															3
1	4	3														7
2	8	13	3													24
3	16	47	25	3												91
4	32	157	147	40	3											379
5	64	503	745	347	58	3										1720
6	128	1573	3483	2480	695	79	3									8441
7	256	4847	15505	15883	6650	1248	103	3								44495
8	512	14797	66867	94920	55783	15386	2072	130	3							250470
9	1024	44903	282265	541467	429618	163485	31962	3242	160	3						1498129
10	2048	135733	1173963	2989600	3119175	1574013	419181	61140	4842	193	3					9479891
11	4096	409247	4831585	16121963	21704650	14137266	4927461	969441	109560	6965	229	3				63222466
12	8192	1231837	19735587	85441400	146349863	120665512	53556954	13652430	2065041	186175	9713	268	3			442902975
13	16384	3703703	80174185	446942587	963540578	991008447	549121144	176428824	34302840	4112966	302731	13197	310	3		3249667899
14	32768	11127493	324400443	2314887120	6228186055	7900599707	5383977599	2136980560	519457224	79545466	7745738	474292	17537	355	3	24907432360

Technical details

Node id: n-32ab10be5b190558 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-3ccc7a2fa091fe08

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:    3;
  1:    4,     3;
  2:    8,    13,      3;
  3:   16,    47,     25,      3;
  4:   32,   157,    147,     40,      3;
  5:   64,   503,    745,    347,     58,      3;
  6:  128,  1573,   3483,   2480,    695,     79,     3;
  7:  256,  4847,  15505,  15883,   6650,   1248,   103,    3;
  8:  512, 14797,  66867,  94920,  55783,  15386,  2072,  130,   3;
  9: 1024, 44903, 282265, 541467, 429618, 163485, 31962, 3242, 160, 3;

%F Formula

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

%t Mathematica

P[0, x_] := 3;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] + (n - 1)*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] = 3; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] + (n - 1)*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 Pascal (1,3) triangle.?
  - 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.