A193727 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 2\,x + 7\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,n x\right)\,P_{n-1}'(x)\)

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

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

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	1														4
2	21	15	2													38
3	147	177	52	4												380
4	1029	1887	926	156	8											4006
5	7203	19041	13960	3880	432	16										44532
6	50421	185775	191642	78360	14240	1136	32									521606
7	352947	1772817	2479612	1401964	370320	47792	2880	64								6428396
8	2470629	16661247	30821366	23184756	8358728	1553104	150304	7104	128							83207366
9	17294403	154892481	372357520	363044560	171750432	43120224	5961984	449792	17152	256						1128888804
10	121060821	1428621135	4405717682	5464294320	3302345600	1076544672	199518144	21369600	1294080	40704	512					16020807270
11	847425747	13099711857	51320136772	79847261524	60463772640	24905950624	5943934080	847797888	72503040	3605760	95232	1024				237352196188
12	5931980229	119592258207	590720928206	1140654673356	1066651112648	544328705888	162746648768	29691623808	3363165696	235150080	9782784	220160	2048			3663926251878
13	41523861603	1088194284321	6737114726680	16009952610040	18281076036432	11386890225392	4180843738368	949017397504	136736058624	12605083392	734435328	25950208	503808	4096		58824714915796
14	290667031221	9876796282095	76284650562122	221603613383880	306236045766560	230139285904528	102209811479776	28291052824320	5042964143360	588599202048	45039920640	2221426688	67518464	1142784	8192	980610816596678

Technical details

Node id: n-2c7f90d092e29857 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-e0ad91a5bacc8439

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,         1;
  2:       21,        15,         2;
  3:      147,       177,        52,         4;
  4:     1029,      1887,       926,       156,         8;
  5:     7203,     19041,     13960,      3880,       432,       16;
  6:    50421,    185775,    191642,     78360,     14240,     1136,      32;
  7:   352947,   1772817,   2479612,   1401964,    370320,    47792,    2880,     64;
  8:  2470629,  16661247,  30821366,  23184756,   8358728,  1553104,  150304,   7104,   128;
  9: 17294403, 154892481, 372357520, 363044560, 171750432, 43120224, 5961984, 449792, 17152, 256;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(2*n-2)*x)*P(n-1,x) + (2*n-2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (2*n - 2)*x)*P[n - 1, x] + (2*n - 2)*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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (2*n - 2)*x)*v[i - 1] + (2*n - 2)*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 Mirror of the triangle A193726.?
  - 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.