A127952 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 16\,x + 8\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 8\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 6 + 18\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	6	18	12													36
3	16	64	80	32												192
4	40	200	360	280	80											960
5	96	576	1344	1536	864	192										4608
6	224	1568	4480	6720	5600	2464	448									21504
7	512	4096	13824	25600	28160	18432	6656	1024								98304
8	1152	10368	40320	88704	120960	104832	56448	17280	2304							442368
9	2560	25600	112640	286720	465920	501760	358400	163840	43520	5120						1966080
10	5632	61952	304128	878592	1655808	2128896	1892352	1148928	456192	107008	11264					8650752
11	12288	147456	798720	2580480	5529600	8257536	8773632	6635520	3502080	1228800	258048	24576				37748736
12	26624	346112	2050048	7321600	17571840	29872128	36900864	33386496	21964800	10250240	3221504	612352	53248			163577856
13	57344	802816	5160960	20185088	53616640	102187008	143818752	151388160	119218176	69386240	29016064	8257536	1433600	114688		704643072
14	122880	1843200	12779520	54312960	158146560	333864960	527155200	632586240	579870720	404152320	210862080	79872000	20766720	3317760	245760	3019898880

Technical details

Node id: n-8528dbef03e9f809 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-efce8cd29f4f1fd0

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:    2,     4;
  2:    6,    18,     12;
  3:   16,    64,     80,     32;
  4:   40,   200,    360,    280,     80;
  5:   96,   576,   1344,   1536,    864,    192;
  6:  224,  1568,   4480,   6720,   5600,   2464,    448;
  7:  512,  4096,  13824,  25600,  28160,  18432,   6656,   1024;
  8: 1152, 10368,  40320,  88704, 120960, 104832,  56448,  17280,  2304;
  9: 2560, 25600, 112640, 286720, 465920, 501760, 358400, 163840, 43520, 5120;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(8*n-16)*x+(4*n-8)*x^2)*P(n-2,x) + (4*n-8+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 18*x + 12*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 6 + 18*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*P[n - 2, x] + (4*n - 8 + (4*n - 8)*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] = 2 + 4*x; v[3] = 6 + 18*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*v[i - 2] + (4*n - 8 + (4*n - 8)*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 Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).?
  - 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.