A395972 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + x + 2\,n x + n^{2} x\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} + 12\,n x + 16\,n x^{2} + 2\,n^{2} x + 16\,n^{2} x^{2} + 4\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 6\,x^{3} - 8\,n x^{3} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 32\,x + 24\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	4	32	24															60
2	8	144	432	192														776
3	16	512	3456	6144	1920													12048
4	32	1600	19200	76800	96000	23040												216672
5	64	4608	86400	614400	1728000	1658880	322560											4414912
6	128	12544	338688	3763200	18816000	40642560	31610880	5160960										100344960
7	256	32768	1204224	19267584	150528000	578027520	1011548160	660602880	92897280									2514108672
8	512	82944	3981312	86704128	975421440	5852528640	18207866880	26754416640	15049359360	1857945600								68788307456
9	1024	204800	12441600	353894400	5419008000	46820229120	227598336000	594542592000	752467968000	371589120000	40874803200							2039678598144
10	2048	495616	37171200	1338163200	26763264000	314735984640	2203151892480	8992456704000	20233027584000	22481141760000	9891702374400	980995276800						65125350672384
11	4096	1179648	107053056	4757913600	120434688000	1849876807680	17625215139840	103593101230080	364194496512000	719396536320000	712202570956800	282526639718400	25505877196800					2227019614720000
12	8192	2768896	299040768	16081747968	502554624000	9769661890560	121578014638080	972624117104640	4923909592842240	15197251829760000	26747163220377600	23873501056204800	8620986492518400	714164561510400				81181467485036544
13	16384	6422528	814055424	52099547136	1970014126080	47280339025920	744665339658240	7780992936837120	53615904455393280	238292908690636800	655305498899251200	1039823601559142400	844856676266803200	279952508112076800	21424936845312000			3141846996378304512
14	32768	14745600	2167603200	162811084800	7326498816000	212761525616640	4137029664768000	54710106587136000	492390959284224000	2978661358632960000	11795498980186521600	29245038793850880000	42242833813340160000	31494657162608640000	9641221580390400000	685597979049984000		128634968016613572608
15	65536	33554432	5662310400	493250150400	26049773568000	900280174510080	21181591883612160	345821908303872000	3939127674273792000	31123971747348480000	167758207718208307200	598938394498066022400	1351770682026885120000	1791696051917291520000	1234076362289971200000	351026165273591808000	23310331287699456000	5554007224762417348608

Technical details

Node id: n-143090c9b4a3eff8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-84f985e00599cd78

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,     32,       24;
   2:    8,    144,      432,        192;
   3:   16,    512,     3456,       6144,        1920;
   4:   32,   1600,    19200,      76800,       96000,        23040;
   5:   64,   4608,    86400,     614400,     1728000,      1658880,        322560;
   6:  128,  12544,   338688,    3763200,    18816000,     40642560,      31610880,       5160960;
   7:  256,  32768,  1204224,   19267584,   150528000,    578027520,    1011548160,     660602880,       92897280;
   8:  512,  82944,  3981312,   86704128,   975421440,   5852528640,   18207866880,   26754416640,    15049359360,     1857945600;
   9: 1024, 204800, 12441600,  353894400,  5419008000,  46820229120,  227598336000,  594542592000,   752467968000,   371589120000,   40874803200;
  10: 2048, 495616, 37171200, 1338163200, 26763264000, 314735984640, 2203151892480, 8992456704000, 20233027584000, 22481141760000, 9891702374400, 980995276800;

%F Formula

The row polynomials satisfy (1+(n^2+2*n+1)*x)*P(n,x) = (2+(2*n^2+12*n+6)*x+(4*n^3+16*n^2+16*n+4)*x^2)*P(n-1,x) - (2*x^2+(2*n^2+8*n+6)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 32*x + 24*x^2.

%t Mathematica

P[1, x_] := 4 + 32*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (2*n^2 + 12*n + 6)*x + (4*n^3 + 16*n^2 + 16*n + 4)*x^2)*P[n - 1, x] - (2*x^2 + (2*n^2 + 8*n + 6)*x^3)*D[P[n - 1, x], x])/(1 + (n^2 + 2*n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 4 + 32*x + 24*x^2; for(n = 2, N, my(i = n); v[i] = ((2 + (2*n^2 + 12*n + 6)*x + (4*n^3 + 16*n^2 + 16*n + 4)*x^2)*v[i - 1] - (2*x^2 + (2*n^2 + 8*n + 6)*x^3)*deriv(v[i - 1], x))/(1 + (n^2 + 2*n + 1)*x)); 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

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.