A096956 → basis A008277

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 6\)

\(P_{1}(x) = 7 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	6															6
1	7	6														13
2	14	25	6													45
3	28	89	49	6												172
4	56	295	285	79	6											721
5	112	941	1435	680	115	6										3289
6	224	2935	6681	4835	1370	157	6									16208
7	448	9029	29659	30856	13055	2469	205	6								85727
8	896	27535	127665	183939	109186	30338	4109	259	6							483933
9	1792	83501	538195	1047360	839055	321552	63210	6440	319	6						2901430
10	3584	252295	2236281	5774995	6081690	3089919	827232	121170	9630	385	6					18397187
11	7168	760469	9197419	31111256	42265135	27711123	9707775	1917762	217470	13865	457	6				122909905
12	14336	2288575	37550145	164753699	284702066	236242996	105373323	26967633	4092462	369985	19349	535	6			862375110
13	28672	6880061	152489155	861318640	1872966095	1938403038	1079229580	348082020	67892253	8162297	602173	26304	619	6		6336080913
14	57344	20668855	616836681	4459082355	12099115210	15441787361	10572239678	4211967760	1027004550	157677520	15388373	944125	34970	709	6	48622805497

Technical details

Node id: n-8501624b7f9f9c89 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-78d59bcf674a8d29

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:    6;
  1:    7,     6;
  2:   14,    25,      6;
  3:   28,    89,     49,       6;
  4:   56,   295,    285,      79,      6;
  5:  112,   941,   1435,     680,    115,      6;
  6:  224,  2935,   6681,    4835,   1370,    157,     6;
  7:  448,  9029,  29659,   30856,  13055,   2469,   205,    6;
  8:  896, 27535, 127665,  183939, 109186,  30338,  4109,  259,   6;
  9: 1792, 83501, 538195, 1047360, 839055, 321552, 63210, 6440, 319, 6;

%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) = 6, P(1,x) = 7 + 6*x.

%t Mathematica

P[0, x_] := 6;
P[1, x_] := 7 + 6*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] = 6; v[2] = 7 + 6*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 (1,6) triangle.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
  - 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.