A159854 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 4\,x + x^{2} - x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 5\,x + 5\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	5	5															11
1	2	16	33	20														71
2	4	48	158	200	87													497
3	8	140	664	1330	1206	409												3757
4	16	404	2616	7418	10564	7439	2066											30523
5	32	1164	9964	37686	75174	82639	47473	11155										265287
6	64	3364	37304	181446	476596	728425	650096	314901	64077									2456273
7	128	9772	138612	846378	2817198	5621533	6936509	5199506	2174687	389946								24134269
8	256	28548	514208	3876490	15932904	39906859	63892214	65928476	42533391	15637264	2504665							250755275
9	512	83852	1911532	17582894	87607270	268145487	535260865	713283059	631264386	357024995	116999102	16923381						2746087335
10	1024	247460	7137384	79429822	473274044	1736355257	4205127004	6951718069	7916037889	6124257797	3080446016	909943187	119928232					31603903185
11	2048	733164	26805028	358740130	2529980958	10966344357	31591178937	63026356834	88688207211	88020510645	60419520809	27341394216	7347364647	888980293				381206119277
12	4096	2179012	101328176	1624072514	13450069920	68117047283	229959602874	542688814524	917849246887	1122196719357	985757298813	607551290318	249701273223	61515021021	6876320041			4807390288059
13	8192	6491980	385644252	7382738182	71364118838	418604323567	1637059076509	4501105508043	8966882089450	13134588445694	14177369139933	11158998866666	6235852823368	2346187996260	533343889525	55382419676		63244513580135
14	16384	19377636	1477534328	33737649174	378874908660	2556180931433	11474258657720	36323906336717	83937500222005	144339157196090	186167785727210	179678407711958	128005082494147	65387058357652	22672575709936	4782627374372	463539664643	866202189870065

Technical details

Node id: n-9320a39650137975 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-5a6e7fab76a29249

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1,     5,       5;
  1:   2,    16,      33,       20;
  2:   4,    48,     158,      200,       87;
  3:   8,   140,     664,     1330,     1206,       409;
  4:  16,   404,    2616,     7418,    10564,      7439,      2066;
  5:  32,  1164,    9964,    37686,    75174,     82639,     47473,     11155;
  6:  64,  3364,   37304,   181446,   476596,    728425,    650096,    314901,     64077;
  7: 128,  9772,  138612,   846378,  2817198,   5621533,   6936509,   5199506,   2174687,    389946;
  8: 256, 28548,  514208,  3876490, 15932904,  39906859,  63892214,  65928476,  42533391,  15637264,   2504665;
  9: 512, 83852, 1911532, 17582894, 87607270, 268145487, 535260865, 713283059, 631264386, 357024995, 116999102, 16923381;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2-(n-4)*x-(2*n-1)*x^2-(n+1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 5*x + 5*x^2.

%t Mathematica

P[0, x_] := 1 + 5*x + 5*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 - (n - 4)*x - (2*n - 1)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 5*x + 5*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 - (n - 4)*x - (2*n - 1)*x^2 - (n + 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A008277 of Riordan array (1-x,x/(1-x)).?
  - Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    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
[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.