A200536 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 21\,x - 72\,x^{2} - 28\,x^{3} + 16\,x^{4} + 63\,n x + 96\,n x^{2} - 28\,n x^{3} - 64\,n x^{4} + 49\,n^{2} x^{2} + 112\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n}(x) = \left(54 + 378\,x + 1098\,x^{2} + 1694\,x^{3} + 1464\,x^{4} + 672\,x^{5} + 128\,x^{6} + 378\,n x + 2046\,n x^{2} + 4396\,n x^{3} + 4680\,n x^{4} + 2464\,n x^{5} + 512\,n x^{6} + 294\,n^{2} x^{2} + 1358\,n^{2} x^{3} + 2344\,n^{2} x^{4} + 1792\,n^{2} x^{5} + 512\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

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	6	56	72																											134
2	36	672	2628	3584	1600																									8520
3	216	6048	39528	108416	146400	96768	25088																							422464
4	1296	48384	443232	1827840	4128400	5483520	4290048	1835008	331776																					18389504
5	7776	362880	4276800	23546880	74604000	149030784	194965120	167444480	91238400	28672000	3964928																			738114048
6	46656	2612736	37651392	259338240	1057752000	2822045184	5179445824	6689292288	6092651520	3842048000	1599848448	396361728	44302336																	28023396352
7	279936	18289152	311568768	2576738304	12856233600	42924639744	101437865088	174981373952	223577335296	211973529600	147494330368	73293889536	24654249984	5035261952	471859200															1021607444480
8	1679616	125411328	2465676288	23800283136	140427302400	563110797312	1628165790720	3515235827712	5783379243264	7323407974400	7147684741120	5339124596736	3000220647424	1228603916288	346344652800	60129542144	4848615424													36107076698112
9	10077696	846526464	18865446912	208182804480	1417987814400	6638084296704	22701283928064	58872276320256	118517522167296	187840536934400	236059591477248	235489105281024	185591146610688	114222399750144	53778500812800	18715319992320	4538304036864	684913065984	48452599808											1245343329943552
10	60466176	5643509760	140583859200	1745725685760	13481157888000	72162430599168	284934136596480	862751159746560	2054461176714240	3911397676032000	6014733793395712	7509883537981440	7620388232970240	6262934344826880	4135098580992000	2162431207931904	875623580958720	264833052180480	56326147276800	7516192768000	473520144384									42111321942523904
11	362797056	37247164416	1025627277312	14153922478080	122198782464000	736177788960768	3296084580569088	11413041294606336	31381747753973760	69761981076480000	126899558640003072	190331026649874432	236319839230418944	243081747395379200	206628791795712000	144283386242924544	81920726164045824	37226965723250688	13219201674117120	3534489649152000	669320724086784	80032420593664	4544075399168							1400926078821728256
12	2176782336	243799621632	7353170731008	111642167476224	1065601466265600	7135364792844288	35720861148758016	139219592779137024	434104348864868352	1103635481105203200	2318196568384806912	4059937654778363904	5963082578081714176	7368035003116290048	7663459730197708800	6697023600899129344	4895103817469657088	2970017979288256512	1478650252310347776	593943081674342400	187760089271107584	44978220373639168	7674943349194752	831230790598656	42949672960000					45969738191129804800
13	13060694016	1584697540608	51955440795648	860666842054656	8999185005158400	66304273331552256	367002138102792192	1589979761401135104	5543813416507711488	15865095924899020800	37792579394813607936	75697343711198576640	128388020545486381056	185237121778556567552	227907728778969907200	239240987531689328640	213961294302562746368	162458582270965972992	104117677325606191104	55834680235760025600	24744703829942992896	8904187026094424064	2536106417237000192	550205514143760384	85426899517440000	8454694661783552	400771988324352			1490907609948483485696
14	78364164096	10239584108544	362590987272192	6507824566763520	73976762551910400	594741212850290688	3606584813537771520	17193768917605023744	66288750426793230336	210884319889263820800	561762193770985439232	1266618066990988787712	2436354950575585148928	4020278721424924213248	5711917916788726579200	7001301718944085704704	7406775339553915666432	6755296357285273534464	5296991623256748326912	3554659153359772057600	2027803286946975842304	974109165051181006848	388924797731387473920	126735022507891359744	32841478166282240000	6510114889573335040	927185994988388352	84477677385285632	3698757115838464	47868544292429890584576

Technical details

Node id: n-51000fd75904f2d1 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-95bcb04e021989f8

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:        6,        56,          72;
  2:       36,       672,        2628,         3584,          1600;
  3:      216,      6048,       39528,       108416,        146400,         96768,          25088;
  4:     1296,     48384,      443232,      1827840,       4128400,       5483520,        4290048,        1835008,          331776;
  5:     7776,    362880,     4276800,     23546880,      74604000,     149030784,      194965120,      167444480,        91238400,        28672000,         3964928;
  6:    46656,   2612736,    37651392,    259338240,    1057752000,    2822045184,     5179445824,     6689292288,      6092651520,      3842048000,      1599848448,       396361728,        44302336;
  7:   279936,  18289152,   311568768,   2576738304,   12856233600,   42924639744,   101437865088,   174981373952,    223577335296,    211973529600,    147494330368,     73293889536,     24654249984,      5035261952,      471859200;
  8:  1679616, 125411328,  2465676288,  23800283136,  140427302400,  563110797312,  1628165790720,  3515235827712,   5783379243264,   7323407974400,   7147684741120,   5339124596736,   3000220647424,   1228603916288,   346344652800,    60129542144,    4848615424;
  9: 10077696, 846526464, 18865446912, 208182804480, 1417987814400, 6638084296704, 22701283928064, 58872276320256, 118517522167296, 187840536934400, 236059591477248, 235489105281024, 185591146610688, 114222399750144, 53778500812800, 18715319992320, 4538304036864, 684913065984, 48452599808;

%F Formula

The row polynomials satisfy (9+(63*n-21)*x+(49*n^2+96*n-72)*x^2+(112*n^2-28*n-28)*x^3+(64*n^2-64*n+16)*x^4)*P(n,x) = (54+(378*n+378)*x+(294*n^2+2046*n+1098)*x^2+(1358*n^2+4396*n+1694)*x^3+(2344*n^2+4680*n+1464)*x^4+(1792*n^2+2464*n+672)*x^5+(512*n^2+512*n+128)*x^6)*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[Together[((54 + (378*n + 378)*x + (294*n^2 + 2046*n + 1098)*x^2 + (1358*n^2 + 4396*n + 1694)*x^3 + (2344*n^2 + 4680*n + 1464)*x^4 + (1792*n^2 + 2464*n + 672)*x^5 + (512*n^2 + 512*n + 128)*x^6)*P[n - 1, x])/(9 + (63*n - 21)*x + (49*n^2 + 96*n - 72)*x^2 + (112*n^2 - 28*n - 28)*x^3 + (64*n^2 - 64*n + 16)*x^4)]];
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] = ((54 + (378*n + 378)*x + (294*n^2 + 2046*n + 1098)*x^2 + (1358*n^2 + 4396*n + 1694)*x^3 + (2344*n^2 + 4680*n + 1464)*x^4 + (1792*n^2 + 2464*n + 672)*x^5 + (512*n^2 + 512*n + 128)*x^6)*v[i - 1])/(9 + (63*n - 21)*x + (49*n^2 + 96*n - 72)*x^2 + (112*n^2 - 28*n - 28)*x^3 + (64*n^2 - 64*n + 16)*x^4)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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

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.