A093562 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 6 + 4\,x\)

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	4														10
2	12	32	16													60
3	24	112	160	64												360
4	48	320	768	768	256											2160
5	96	832	2816	4608	3584	1024										12960
6	192	2048	8960	20480	25600	16384	4096									77760
7	384	4864	26112	76800	133120	135168	73728	16384								466560
8	768	11264	71680	258048	573440	802816	688128	327680	65536							2799360
9	1536	25600	188416	802816	2179072	3899392	4587520	3407872	1441792	262144						16796160
10	3072	57344	479232	2359296	7569408	16515072	24772608	25165824	16515072	6291456	1048576					100776960
11	6144	126976	1187840	6635520	24576000	63307776	115605504	149422080	133693440	78643200	27262976	4194304				604661760
12	12288	278528	2883584	18022400	75694080	224919552	484442112	761266176	865075200	692060160	369098752	117440512	16777216			3627970560
13	24576	606208	6881280	47579136	223477760	752615424	1868562432	3460300800	4775215104	4844421120	3506438144	1711276032	503316480	67108864		21767823360
14	49152	1310720	16187392	122683392	637272064	2399141888	6747586560	14394851328	23391633408	28789702656	26390560768	17448304640	7851737088	2147483648	268435456	130606940160

Technical details

Node id: n-5d201923e61d3148 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-a9636442fdb13e67

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,     4;
  2:   12,    32,     16;
  3:   24,   112,    160,     64;
  4:   48,   320,    768,    768,     256;
  5:   96,   832,   2816,   4608,    3584,    1024;
  6:  192,  2048,   8960,  20480,   25600,   16384,    4096;
  7:  384,  4864,  26112,  76800,  133120,  135168,   73728,   16384;
  8:  768, 11264,  71680, 258048,  573440,  802816,  688128,  327680,   65536;
  9: 1536, 25600, 188416, 802816, 2179072, 3899392, 4587520, 3407872, 1441792, 262144;

%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) = 1, P(1,x) = 6 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 4*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] = 1; v[2] = 6 + 4*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 (5,1) Pascal 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

Related ?

Same fingerprint ?

A156319 → basis A013609

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.