A096956 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 7 + 24\,x\)

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	24														31
2	14	76	96													186
3	28	208	496	384												1116
4	56	528	1824	2752	1536											6696
5	112	1280	5760	12800	14080	6144										40176
6	224	3008	16640	48640	79360	68608	24576									241056
7	448	6912	45312	163840	353280	454656	323584	98304								1446336
8	896	15616	118272	508928	1361920	2322432	2465792	1490944	393216							8678016
9	1792	34816	299008	1490944	4759552	10092544	14221312	12845056	6750208	1572864						52068096
10	3584	76800	737280	4177920	15482880	39223296	68812800	82575360	64880640	30146560	6291456					312408576
11	7168	167936	1781760	11304960	47677440	140378112	294518784	440401920	460062720	319815680	133169152	25165824				1874451456
12	14336	364544	4235264	29736960	140574720	471465984	1150550016	2058878976	2681733120	2479882240	1545601024	583008256	100663296			11246708736
13	28672	786432	9928704	76414976	400097280	1505230848	4186963968	8719958016	13598982144	15686696960	13010731008	7348420608	2533359616	402653184		67480252416
14	57344	1687552	23003136	192544768	1105854464	4610850816	14394851328	34187771904	62077796352	85769322496	88768249856	66739765248	34460401664	10938744832	1610612736	404881514496

Technical details

Node id: n-2727b5676a7ef22c · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e2877f1d1284fed3

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,    24;
  2:   14,    76,     96;
  3:   28,   208,    496,     384;
  4:   56,   528,   1824,    2752,    1536;
  5:  112,  1280,   5760,   12800,   14080,     6144;
  6:  224,  3008,  16640,   48640,   79360,    68608,    24576;
  7:  448,  6912,  45312,  163840,  353280,   454656,   323584,    98304;
  8:  896, 15616, 118272,  508928, 1361920,  2322432,  2465792,  1490944,  393216;
  9: 1792, 34816, 299008, 1490944, 4759552, 10092544, 14221312, 12845056, 6750208, 1572864;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 6, P(1,x) = 7 + 24*x.

%t Mathematica

P[0, x_] := 6;
P[1, x_] := 7 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (4*n - 4)*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 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (4*n - 4)*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.