A096956 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

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	19	6													39
3	28	52	31	6												117
4	56	132	114	43	6											351
5	112	320	360	200	55	6										1053
6	224	752	1040	760	310	67	6									3159
7	448	1728	2832	2560	1380	444	79	6								9477
8	896	3904	7392	7952	5320	2268	602	91	6							28431
9	1792	8704	18688	23296	18592	9856	3472	784	103	6						85293
10	3584	19200	46080	65280	60480	38304	16800	5040	990	115	6					255879
11	7168	41984	111360	176640	186240	137088	71904	26880	7020	1220	127	6				767637
12	14336	91136	264704	464640	549120	460416	280896	125664	40920	9460	1474	139	6			2302911
13	28672	196608	620544	1193984	1562880	1469952	1022208	532224	207504	59840	12408	1752	151	6		6908733
14	57344	421888	1437696	3008512	4319744	4502784	3514368	2086656	947232	327184	84656	15912	2054	163	6	20726199

Technical details

Node id: n-837649989bd08cfc · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-81eb69734bdba80b

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): yes

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,   19,     6;
  3:   28,   52,    31,     6;
  4:   56,  132,   114,    43,     6;
  5:  112,  320,   360,   200,    55,    6;
  6:  224,  752,  1040,   760,   310,   67,    6;
  7:  448, 1728,  2832,  2560,  1380,  444,   79,   6;
  8:  896, 3904,  7392,  7952,  5320, 2268,  602,  91,   6;
  9: 1792, 8704, 18688, 23296, 18592, 9856, 3472, 784, 103, 6;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*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)]];
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)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.