A134239 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 8\,x + 2\,x^{2} - 4\,n - 4\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 4\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 6 + 2\,x\)

\(P_{2}(x) = 18 + 15\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	2														8
2	18	15	3													36
3	48	64	28	4												144
4	120	220	150	45	5											540
5	288	672	624	288	66	6										1944
6	672	1904	2240	1400	490	91	7									6804
7	1536	5120	7296	5760	2720	768	120	8								23328
8	3456	13248	22176	21168	12600	4788	1134	153	9							78732
9	7680	33280	64000	71680	51520	24640	7840	1600	190	10						262440
10	16896	81664	177408	228096	192192	110880	44352	12144	2178	231	11					866052
11	36864	196608	476160	691200	668160	451584	217728	74880	18000	2880	276	12				2834352
12	79872	465920	1244672	2013440	2196480	1702272	960960	398112	120120	25740	3718	325	13			9211644
13	172032	1089536	3182592	5677056	6899200	6031872	3902976	1892352	687456	184800	35728	4704	378	14		29760696
14	368640	2519040	7987200	15575040	20866560	20317440	14826240	8236800	3500640	1132560	274560	48360	5850	435	15	95659380

Technical details

Node id: n-de4fe2266f386481 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-1c35c8e273da035a

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:    1;
  1:    6,     2;
  2:   18,    15,     3;
  3:   48,    64,    28,     4;
  4:  120,   220,   150,    45,     5;
  5:  288,   672,   624,   288,    66,     6;
  6:  672,  1904,  2240,  1400,   490,    91,    7;
  7: 1536,  5120,  7296,  5760,  2720,   768,  120,    8;
  8: 3456, 13248, 22176, 21168, 12600,  4788, 1134,  153,   9;
  9: 7680, 33280, 64000, 71680, 51520, 24640, 7840, 1600, 190, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 2*x, P(2,x) = 18 + 15*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 2*x;
P[2, x_] := 18 + 15*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];
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] = 6 + 2*x; v[3] = 18 + 15*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (4*n - 8 + (2*n - 4)*x)*v[i - 1])/(n - 2)); 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 A127899(unsigned) * A007318.?
  - 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 for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[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.