A200536 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 11\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)

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

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	15	11	2																											28
2	225	330	181	44	4																									784
3	3375	7425	6795	3311	906	132	8																							21952
4	50625	148500	190350	139260	63601	18568	3384	352	16																					614656
5	759375	2784375	4590000	4479750	2866575	1256651	382210	79640	10880	880	32																			17210368
6	11390625	50118750	100996875	123255000	101455875	59341590	25289461	7912212	1803660	292160	31920	2112	64																	481890304
7	170859375	877078125	2089040625	3060028125	3079636875	2252648475	1235011155	515550431	164668154	40047084	7299880	967120	88032	4928	128															13492928512
8	2562890625	15035625000	41325187500	70634025000	84032943750	73785789000	49463574300	25823676120	10611099361	3443156816	879352432	174899648	26558560	2976512	232192	11264	256													377801998336
9	38443359375	253726171875	790395468750	1544158687500	2120118806250	2172417266250	1721663181000	1079026037100	542154076335	220016797451	72287210178	19182685104	4080979392	686591136	89341632	8676096	592128	25344	512											10578455953408
10	576650390625	4228769531250	14723806640625	32364182812500	50368318593750	58995883237500	53961775256250	39468520080000	23444923915125	11421998875650	4588801077301	1522933183420	416798647380	93555010560	17054585760	2486081664	283002240	24245760	1470720	56320	1024									296196766695424
11	8649755859375	69774697265625	268526865234375	655882154296875	1140978403125000	1503718118718750	1559117981643750	1303599095493750	893751130119375	508161186361125	241363851621915	96164807352911	32181846882922	9033976646420	2118521197320	412001689440	65701103424	8448874368	854768640	65514240	3576320	123904	2048							8293509467471872
12	129746337890625	1141767773437500	4812724160156250	12931577226562500	24866433474609375	36418298523750000	42209625836812500	39711720467925000	30864092965509375	20060878417717500	10997733084539850	5113796850856980	2023268287369681	681839580114264	195515254836264	47551711804960	9754577529840	1673448632064	237161354496	27282917376	2483838720	172226560	8546304	270336	4096					232218265089212416
13	1946195068359375	18553726318359375	85009800585937500	249197159707031250	524869299931640625	845668400530078125	1083478538262656250	1132818288271312500	984209571303440625	719841639822215625	447363844794009000	237803773528228050	108596255839051695	42711138564494411	14479500778540226	4227622640501832	1060418002474688	227505505919120	41474510329824	6364915924224	811692380928	84471459072	6990364160	442517504	20127744	585728	8192			6502111422497947648
14	29192926025390625	299714040527343750	1483130388427734375	4710172654687500000	10784227856923828125	18956982626613281250	26604269079633984375	30601875046019062500	29391101817061359375	23889566458213706250	16597134852561388125	9927742175301951000	5139513035984301975	2310833439753440910	904207547565645301	308111125300458788	91369120639720924	23532425897012032	5245514225007056	1006704364329664	165138481540032	22925519924736	2657426274048	252474686464	19150337024	1115226112	46821376	1261568	16384	182059119829942534144

Technical details

Node id: n-f2085d6c0d79c3b9 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-8e69101448399e5f

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:          15,           11,            2;
  2:         225,          330,          181,            44,             4;
  3:        3375,         7425,         6795,          3311,           906,           132,             8;
  4:       50625,       148500,       190350,        139260,         63601,         18568,          3384,           352,           16;
  5:      759375,      2784375,      4590000,       4479750,       2866575,       1256651,        382210,         79640,        10880,          880,          32;
  6:    11390625,     50118750,    100996875,     123255000,     101455875,      59341590,      25289461,       7912212,      1803660,       292160,       31920,        2112,         64;
  7:   170859375,    877078125,   2089040625,    3060028125,    3079636875,    2252648475,    1235011155,     515550431,    164668154,     40047084,     7299880,      967120,      88032,      4928,      128;
  8:  2562890625,  15035625000,  41325187500,   70634025000,   84032943750,   73785789000,   49463574300,   25823676120,  10611099361,   3443156816,   879352432,   174899648,   26558560,   2976512,   232192,   11264,    256;
  9: 38443359375, 253726171875, 790395468750, 1544158687500, 2120118806250, 2172417266250, 1721663181000, 1079026037100, 542154076335, 220016797451, 72287210178, 19182685104, 4080979392, 686591136, 89341632, 8676096, 592128, 25344, 512;

%F Formula

The row polynomials satisfy P(n,x) = (15+11*x+2*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[(15 + 11*x + 2*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] = (15 + 11*x + 2*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_A007318 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 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 (above)
    
    ↑ Iterated real-rootedness preservers (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A007318 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)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A200536 → basis A013610 → x ↦ x/(x+1)

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.