A200536 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 14\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	8	14	6																											28
2	64	224	292	168	36																									784
3	512	2688	5856	6776	4392	1512	216																							21952
4	4096	28672	87552	152320	165136	114240	49248	12096	1296																					614656
5	32768	286720	1126400	2616320	3978880	4139744	2984160	1471680	475200	90720	7776																			17210368
6	262144	2752512	13221888	38420480	75217920	104520192	105702976	78390144	42310080	16208640	4183488	653184	46656																	481890304
7	2097152	25690112	145883136	508985344	1218961408	2119735296	2760214016	2734083968	2070160512	1192351104	514249344	161046144	34618752	4572288	279936															13492928512
8	16777216	234881024	1539309568	6268387328	17752784896	37077254144	59071774720	73234079744	71399743744	54925559808	33227873280	15641966592	5617092096	1487517696	273964032	31352832	1679616													377801998336
9	134217728	2113929216	15703474176	73106718720	239015559168	582767345664	1098172465152	1635341008896	1950905714688	1878405369344	1463179286016	919879317504	463291508736	184391230464	56719512576	13011425280	2096160768	211631616	10077696											10578455953408
10	1073741824	18790481920	156028108800	817385963520	3029839380480	8446996905984	18378215915520	31953746657280	45091054632960	52151969013760	49708543747072	39113976760320	25363718231040	13480486871040	5814982379520	2004511961088	539246315520	109107855360	15620428800	1410877440	60466176									296196766695424
11	8589934592	165356240896	1517734068224	8836224122880	36618287185920	114898042355712	283462720290816	563606977511424	918350185758720	1240212996915200	1398342243966976	1321743240624128	1048756682975232	697619810764800	387428984616960	178328770228224	67267032256512	20449383026688	4887951298560	884620154880	113958586368	9311791104	362797056							8293509467471872
12	68719476736	1443109011456	14508399525888	92930207383552	425759839617024	1484857704185856	4095984078422016	9166722158297088	16938075492974592	26160248441012224	34059821023100928	37592015322021888	35284512296341504	28194011491516416	19158649325494272	11036354811052032	5359312948948992	2175306137174016	728997166301184	198204577579008	42624058650624	6977635467264	817018970112	60949905408	2176782336					232218265089212416
13	549755813888	12506944765952	136683039227904	955217906499584	4794152017461248	18397080632426496	56110439523680256	139586700589400064	288414618630488064	501415377379524608	740350499316826112	934535107545661440	1012923239017644032	945087356013043712	759692429263233024	525675997994434560	312335366899286016	158650959248990208	68442140944539648	24843433771892736	7489839553118208	1841785370320896	359967400206336	53791677628416	5772826755072	396174385152	13060694016			6502111422497947648
14	4398046511104	107752139522048	1271860075429888	9630347469783040	52546365066051584	220026080742866944	735207557148180480	2012622241841283072	4598193394437586944	8886647883399430144	14673106989630881792	20849680115078004736	25628980413681369088	27348834839625334784	25386301852394340352	20511626129719001088	14416301482695770112	8795958798548533248	4642662758437896192	2108843198892638208	818379634898681856	268652639337578496	73603771704852480	16520589245841408	2959070502076416	406739035422720	40287887474688	2559896027136	78364164096	182059119829942534144

Technical details

Node id: n-feafdaba4f567b18 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-a0b8814ff2967252

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:         8,         14,           6;
  2:        64,        224,         292,         168,           36;
  3:       512,       2688,        5856,        6776,         4392,         1512,           216;
  4:      4096,      28672,       87552,      152320,       165136,       114240,         49248,         12096,          1296;
  5:     32768,     286720,     1126400,     2616320,      3978880,      4139744,       2984160,       1471680,        475200,         90720,          7776;
  6:    262144,    2752512,    13221888,    38420480,     75217920,    104520192,     105702976,      78390144,      42310080,      16208640,       4183488,       653184,        46656;
  7:   2097152,   25690112,   145883136,   508985344,   1218961408,   2119735296,    2760214016,    2734083968,    2070160512,    1192351104,     514249344,    161046144,     34618752,      4572288,      279936;
  8:  16777216,  234881024,  1539309568,  6268387328,  17752784896,  37077254144,   59071774720,   73234079744,   71399743744,   54925559808,   33227873280,  15641966592,   5617092096,   1487517696,   273964032,    31352832,    1679616;
  9: 134217728, 2113929216, 15703474176, 73106718720, 239015559168, 582767345664, 1098172465152, 1635341008896, 1950905714688, 1878405369344, 1463179286016, 919879317504, 463291508736, 184391230464, 56719512576, 13011425280, 2096160768, 211631616, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (8+14*x+6*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(8 + 14*x + 6*x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (8 + 14*x + 6*x^2)*v[i - 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? 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 ?

A200536 → 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.