A200536 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(28 + 30\,x + 8\,x^{2}\right)\,P_{n-1}(x)\)

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

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	28	30	8																											66
2	784	1680	1348	480	64																									4356
3	21952	70560	94416	67320	26976	5760	512																							287496
4	614656	2634240	4936064	5281920	3530256	1509120	402944	61440	4096																					18974736
5	17210368	92198400	222154240	317049600	296793280	190418400	84798080	25881600	5181440	614400	32768																			1252332576
6	481890304	3097866240	9123953664	16279603200	19598933760	16771910400	10461244480	4791974400	1599912960	379699200	60801024	5898240	262144																	82653950016
7	13492928512	101196963840	352261812224	754330429440	1110149870592	1187818329600	952863627520	582187900800	272246750720	96964761600	25892708352	5026775040	670695424	55050240	2097152															5455160701056
8	377801998336	3238302842880	13007183085568	32498682101760	56532203757568	72598052782080	71195930423296	54389716684800	32711455064320	15539919052800	5811912687616	1693249044480	376724389888	61876469760	7075790848	503316480	16777216													360040606269696
9	10578455953408	102006539550720	464372627668992	1326085014159360	2661919632949248	3988701047439360	4623685265375232	4239574402129920	3117179685731328	1851579118886400	890622767351808	346087706296320	107841055752192	26580265205760	5068211355648	721378344960	72175583232	4529848320	134217728											23762680013799936
10	296196766695424	3173536786022400	16147257408880640	51877611542937600	118031281168711680	202149898430054400	270419575917281280	289328249600409600	251457745387376640	179276201117798400	105422248538293248	51221771747942400	20527162888765440	6748180748697600	1802046320148480	384887055974400	64207957524480	8063130009600	717058211840	40265318400	1073741824									1568336880910795776
11	8293509467471872	97744933009489920	549698885162893312	1962379139756851200	4990382278283100160	9616156483446374400	14580495327935201280	17830977453770342400	17884020966197084160	14878091989722931200	10341770955705176064	6031086674033571840	2954791701630050304	1214538121610035200	417120022535208960	118823673161318400	27760805051105280	5231102813798400	775635190415360	87144215347200	6974490017792	354334801920	8589934592							103510234140112521216
12	232218265089212416	2985663408289873920	18390264850585485312	72219541932154552320	202999669065935486976	434662883003046297600	736661621911841669120	1013611480411196620800	1152325873290110238720	1095755024328317337600	878984514181009539072	598148291461878743040	346400935512289972224	170899511846251069440	71753837892327309312	25556968497453465600	7678972916552171520	1929884415610060800	400737310154424320	67557916252569600	9014693704237056	916309797765120	66666482368512	3092376453120	68719476736					6831675453247426400256
13	6502111422497947648	90565123384792842240	606355064185803505664	2597740426884211015680	7997699110615514087424	18838307131520597360640	35290409256150439559168	53958273172893125836800	68566761839753718661120	73359808723185765580800	66702824113238669131776	51883727780989429678080	34675551051648557826048	19962400728758759055360	9907300300471016521728	4235406349468524871680	1555750999725683245056	488861699113274572800	130548960485043077120	29352816284585164800	5485047392531120128	836560815485091840	101473505073168384	9417059393863680	628027297890304	26800595927040	549755813888			450890579914330142416896
14	182059119829942534144	2730886797449138012160	19746912390126267006976	91651904865410356346880	306718628417247152963584	788185496416115836846080	1617262266002754341175296	2720250383577685489090560	3820940800749101414088704	4542743684824958002790400	4617007431584285452402688	4040707571050350229585920	3061049855781751862525952	2014283574202617425756160	1153680838689139696877568	575509592629319264501760	249881620880143009185792	94243908362690384363520	30767229864784909303808	8649241263425873510400	2078557499408770924544	422797654886197493760	71818461746850430976	10000369339103969280	1111884586742185984	94927710773575680	5843629423722496	230897441832960	4398046511104	29758778274345789399515136

Technical details

Node id: n-86e0f3639d6a7bfe · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4305ea87645379bc

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:             28,              30,               8;
  2:            784,            1680,            1348,              480,               64;
  3:          21952,           70560,           94416,            67320,            26976,             5760,              512;
  4:         614656,         2634240,         4936064,          5281920,          3530256,          1509120,           402944,            61440,             4096;
  5:       17210368,        92198400,       222154240,        317049600,        296793280,        190418400,         84798080,         25881600,          5181440,           614400,           32768;
  6:      481890304,      3097866240,      9123953664,      16279603200,      19598933760,      16771910400,      10461244480,       4791974400,       1599912960,        379699200,        60801024,         5898240,          262144;
  7:    13492928512,    101196963840,    352261812224,     754330429440,    1110149870592,    1187818329600,     952863627520,     582187900800,     272246750720,      96964761600,     25892708352,      5026775040,       670695424,       55050240,       2097152;
  8:   377801998336,   3238302842880,  13007183085568,   32498682101760,   56532203757568,   72598052782080,   71195930423296,   54389716684800,   32711455064320,   15539919052800,   5811912687616,   1693249044480,    376724389888,    61876469760,    7075790848,    503316480,    16777216;
  9: 10578455953408, 102006539550720, 464372627668992, 1326085014159360, 2661919632949248, 3988701047439360, 4623685265375232, 4239574402129920, 3117179685731328, 1851579118886400, 890622767351808, 346087706296320, 107841055752192, 26580265205760, 5068211355648, 721378344960, 72175583232, 4529848320, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (28+30*x+8*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[(28 + 30*x + 8*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] = (28 + 30*x + 8*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 (−∞, -1]
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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (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
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) (above)

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.