A395972 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + x + x^{2} + 4\,n x + 4\,n x^{2} + 3\,n^{2} x + 6\,n^{2} x^{2} + 4\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(2 + 10\,x + 10\,x^{2} + 8\,x^{3} + 24\,n x + 56\,n x^{2} + 48\,n x^{3} + 6\,n^{2} x + 76\,n^{2} x^{2} + 94\,n^{2} x^{3} + 28\,n^{3} x^{2} + 80\,n^{3} x^{3} + 2\,n^{4} x^{2} + 30\,n^{4} x^{3} + 4\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 12\,x^{3} - 16\,x^{4} - 20\,n x^{3} - 44\,n x^{4} - 6\,n^{2} x^{3} - 42\,n^{2} x^{4} - 16\,n^{3} x^{4} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 64\,x + 72\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	4	64	72															140
2	8	288	1296	768														2360
3	16	1024	10368	24576	9600													45584
4	32	3200	57600	307200	480000	138240												986272
5	64	9216	259200	2457600	8640000	9953280	2257920											23577280
6	128	25088	1016064	15052800	94080000	243855360	221276160	41287680										616593280
7	256	65536	3612672	77070336	752640000	3468165120	7080837120	5284823040	836075520									17503289600
8	512	165888	11943936	346816512	4877107200	35115171840	127455068160	214035333120	135444234240	18579456000								535865297408
9	1024	409600	37324800	1415577600	27095040000	280921374720	1593188352000	4756340736000	6772211712000	3715891200000	449622835200							17596724562944
10	2048	991232	111513600	5352652800	133816320000	1888415907840	15422063247360	71939653632000	182097248256000	224811417600000	108808726118400	11771943321600						616878749562880
11	4096	2359296	321159168	19031654400	602173440000	11099260846080	123376505978880	828744809840640	3277750468608000	7193965363200000	7834228280524800	3390319676620800	331576403558400					22991682297794560
12	8192	5537792	897122304	64326991872	2512773120000	58617971343360	851046102466560	7780992936837120	44315186335580160	151972518297600000	294218795424153600	286482012674457600	112072824402739200	9998303861145600				907752876009103360
13	16384	12845056	2442166272	208398188544	9850070630400	283682034155520	5212657377607680	62247943494696960	482543140098539520	2382929086906368000	7208360487891763200	12477883218709708800	10983136791468441600	3919335113569075200	321374052679680000			37843316235153883136
14	32768	29491200	6502809600	651244339200	36632494080000	1276569153699840	28959207653376000	437680852697088000	4431518633558016000	29786613586329600000	129750488782051737600	350940465526210560000	549156839573422080000	440925200276520960000	144618323705856000000	10969567664799744000		1661046971668523614208
15	65536	67108864	16986931200	1973000601600	130248867840000	5401681047060480	148271143185285120	2766575266430976000	35452149068464128000	311239717473484800000	1845340284900291379200	7187260733976792268800	17573018866349506560000	25083744726842081280000	18511145434349568000000	5616418644377468928000	396275631890890752000	76562816569558133964800

Technical details

Node id: n-b947e3321c316ff1 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-53219bd9184fb09a

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    4,     64,        72;
   2:    8,    288,      1296,        768;
   3:   16,   1024,     10368,      24576,         9600;
   4:   32,   3200,     57600,     307200,       480000,        138240;
   5:   64,   9216,    259200,    2457600,      8640000,       9953280,        2257920;
   6:  128,  25088,   1016064,   15052800,     94080000,     243855360,      221276160,       41287680;
   7:  256,  65536,   3612672,   77070336,    752640000,    3468165120,     7080837120,     5284823040,       836075520;
   8:  512, 165888,  11943936,  346816512,   4877107200,   35115171840,   127455068160,   214035333120,    135444234240,     18579456000;
   9: 1024, 409600,  37324800, 1415577600,  27095040000,  280921374720,  1593188352000,  4756340736000,   6772211712000,   3715891200000,    449622835200;
  10: 2048, 991232, 111513600, 5352652800, 133816320000, 1888415907840, 15422063247360, 71939653632000, 182097248256000, 224811417600000, 108808726118400, 11771943321600;

%F Formula

The row polynomials satisfy (1+(3*n^2+4*n+1)*x+(n^4+4*n^3+6*n^2+4*n+1)*x^2)*P(n,x) = (2+(6*n^2+24*n+10)*x+(2*n^4+28*n^3+76*n^2+56*n+10)*x^2+(4*n^5+30*n^4+80*n^3+94*n^2+48*n+8)*x^3)*P(n-1,x) - (2*x^2+(6*n^2+20*n+12)*x^3+(2*n^4+16*n^3+42*n^2+44*n+16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 64*x + 72*x^2.

%t Mathematica

P[1, x_] := 4 + 64*x + 72*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n^2 + 24*n + 10)*x + (2*n^4 + 28*n^3 + 76*n^2 + 56*n + 10)*x^2 + (4*n^5 + 30*n^4 + 80*n^3 + 94*n^2 + 48*n + 8)*x^3)*P[n - 1, x] - (2*x^2 + (6*n^2 + 20*n + 12)*x^3 + (2*n^4 + 16*n^3 + 42*n^2 + 44*n + 16)*x^4)*D[P[n - 1, x], x])/(1 + (3*n^2 + 4*n + 1)*x + (n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 4 + 64*x + 72*x^2; for(n = 2, N, my(i = n); v[i] = ((2 + (6*n^2 + 24*n + 10)*x + (2*n^4 + 28*n^3 + 76*n^2 + 56*n + 10)*x^2 + (4*n^5 + 30*n^4 + 80*n^3 + 94*n^2 + 48*n + 8)*x^3)*v[i - 1] - (2*x^2 + (6*n^2 + 20*n + 12)*x^3 + (2*n^4 + 16*n^3 + 42*n^2 + 44*n + 16)*x^4)*deriv(v[i - 1], x))/(1 + (3*n^2 + 4*n + 1)*x + (n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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 Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 ?

A144084 → (θ+1)²

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.