A272866 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 15\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)

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

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	9	15	5																											29
2	81	270	315	150	25																									841
3	729	3645	7290	7425	4050	1125	125																							24389
4	6561	43740	123930	194400	184275	108000	38250	7500	625																					707281
5	59049	492075	1804275	3827250	5194125	4708125	2885625	1181250	309375	46875	3125																			20511149
6	531441	5314410	23914845	63969750	113177250	139421250	122563125	77456250	34931250	10968750	2278125	281250	15625																	594823321
7	4782969	55801305	297606960	961022475	2097715725	3272298750	3760273125	3232659375	2089040625	1009968750	359690625	91546875	15750000	1640625	78125															17249876309
8	43046721	573956280	3539397060	13392313200	34782813450	65721537000	93415518000	101859525000	86092621875	56588625000	28831950000	11269125000	3313406250	708750000	104062500	9375000	390625													500246412961
9	387420489	5811307335	40679151345	176491556100	531627004350	1180197600750	2000476784250	2646576180000	2769804061875	2309984578125	1538780034375	816844500000	343017281250	112425468750	28134843750	5189062500	664453125	52734375	1953125											14507145975869
10	3486784401	58113073350	455219074575	2227667811750	7635412137375	19478641252500	38365390091250	59727325387500	74629263178125	75569803031250	62347809290625	41983223906250	23033723203125	10241310937500	3654682031250	1030851562500	224490234375	36386718750	4130859375	292968750	9765625									420707233300201
11	31381059609	575319426165	4986101693430	27167861791125	104409821785500	300977292391875	675645190295625	1210419986118750	1759400199871875	2098203801890625	2067823644975000	1690915169671875	1148790913875000	647593766015625	301680418359375	115304449218750	35756595703125	8849091796875	1705429687500	246533203125	25136718750	1611328125	48828125							12200509765705829
12	282429536481	5648590729620	53661611931390	322178878652400	1372136831403525	4410782267265000	11117515207466250	22533344191462500	37369127542106250	51326937145687500	58880470832493750	56726610211125000	46041963994828125	31514783450625000	18172984824843750	8800915148437500	3559778191406250	1192513007812500	326868222656250	72045703125000	12451376953125	1624218750000	150292968750	8789062500	244140625					353814783205469041
13	2541865828329	55073759613795	569095516009215	3732777040490550	17450222722074675	61889987269699875	173080055033188875	391616737171481250	729909886788225000	1135146068400093750	1486673932388287500	1650381240115968750	1559679183282796875	1257895562033671875	866488435157109375	509376925961718750	254916655073437500	108133865683593750	38628400078125000	11513999707031250	2827089052734375	561617138671875	87972802734375	10454589843750	885498046875	47607421875	1220703125			10260628712958602189
14	22876792454961	533791823949090	5960675367431505	42406794902622150	215889137686076400	837427111460871750	2573321417954571375	6430201396389663750	13308840313832188125	23123046603281625000	34056805851437118750	42829270488868500000	46226200913226140625	42968154008124843750	34465225263333046875	23871196671180468750	14267345960872265625	7343839247062500000	3244246861324218750	1223721326953125000	391295797470703125	105030888574218750	23351457568359375	4221769042968750	604652343750000	65983886718750	5152587890625	256347656250	6103515625	297558232675799463481

Technical details

Node id: n-2406581cbd276ab3 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-970ef418506da5d4

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         9,         15,           5;
  2:        81,        270,         315,          150,           25;
  3:       729,       3645,        7290,         7425,         4050,          1125,           125;
  4:      6561,      43740,      123930,       194400,       184275,        108000,         38250,          7500,           625;
  5:     59049,     492075,     1804275,      3827250,      5194125,       4708125,       2885625,       1181250,        309375,         46875,          3125;
  6:    531441,    5314410,    23914845,     63969750,    113177250,     139421250,     122563125,      77456250,      34931250,      10968750,       2278125,       281250,        15625;
  7:   4782969,   55801305,   297606960,    961022475,   2097715725,    3272298750,    3760273125,    3232659375,    2089040625,    1009968750,     359690625,     91546875,     15750000,      1640625,       78125;
  8:  43046721,  573956280,  3539397060,  13392313200,  34782813450,   65721537000,   93415518000,  101859525000,   86092621875,   56588625000,   28831950000,  11269125000,   3313406250,    708750000,   104062500,    9375000,    390625;
  9: 387420489, 5811307335, 40679151345, 176491556100, 531627004350, 1180197600750, 2000476784250, 2646576180000, 2769804061875, 2309984578125, 1538780034375, 816844500000, 343017281250, 112425468750, 28134843750, 5189062500, 664453125, 52734375, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (9+15*x+5*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[(9 + 15*x + 5*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] = (9 + 15*x + 5*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_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272866 → basis A013610

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.