A257610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 6\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-4 - 208\,x - 48\,x^{2} + 6\,n + 252\,n x + 216\,n x^{2} + 18\,n^{2} + 216\,n^{2} x + 81\,n^{2} x^{2} - 216\,n^{3} x - 1134\,n^{3} x^{2} + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(69\,x - 218\,x^{2} - 71\,x^{3} - 36\,n x + 117\,n x^{2} + 276\,n x^{3} - 108\,n^{2} x + 324\,n^{2} x^{2} + 297\,n^{2} x^{3} + 216\,n^{3} x^{2} - 1566\,n^{3} x^{3} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(45\,x^{2} - 30\,x^{3} - 15\,x^{4} - 27\,n x^{2} - 27\,n x^{3} + 54\,n x^{4} - 81\,n^{2} x^{2} + 81\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(4 - 4\,x - 15\,n + 36\,n x + 9\,n^{2} - 108\,n^{2} x + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(-12\,x - 8\,x^{2} - 9\,n x + 36\,n x^{2} + 54\,n^{2} x - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	4	80	36													120
3	8	528	1188	128												1852
4	16	2992	19008	11968	400											34384
5	32	15856	226116	401984	99100	1152										744240
6	64	81456	2308392	8843648	6412200	733104	3136									18382000
7	128	412400	21582828	154733440	241771000	86331312	5051900	8192								509891200
8	256	2073776	191220624	2354348864	6769588000	5297284944	1041090064	33180416	20736							15688807680
9	512	10395504	1637082900	32696793024	157311623700	226528738128	100133928636	11641478400	210508956	51200						530170600960
10	1024	52036912	13704800184	426603224064	3224137513800	7701977096352	6319309527048	1706412896256	123343201656	1300922800	123904					19516841344000
11	2048	260315632	113033809980	5326101784320	60468978558000	223788771035424	304601382798216	154800585108480	26963390283120	1255931222000	7874547868	294912				777326309760000
12	4096	1301864880	923013205344	64414640497600	1062938584800000	5807141677443168	12184055311928640	10323807426565632	3443921014752000	402591503110000	12409399760736	46867135680	692224			33302250741760000
13	8192	6509946992	7486627502052	761070241155392	17799928584747500	138517829131445856	425476902640751976	555724934061390336	311665115545753176	71199714338990000	5755593698737652	119786040032832	275045260412	1605632		1527028642467328000
14	16384	32551078448	60449620484232	8837607252836992	287252512244234600	3098128112933635152	13411530117077873424	25563346966823955456	22170080397618525456	8605911424815653200	1390302159262095464	79538465275532928	1135109540203912	1595002843952	3686400	74616124950022656000

Technical details

Node id: n-f882fbd3dc52436c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-344812a88580a3fd

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:   2,        8;
  2:   4,       80,         36;
  3:   8,      528,       1188,         128;
  4:  16,     2992,      19008,       11968,          400;
  5:  32,    15856,     226116,      401984,        99100,         1152;
  6:  64,    81456,    2308392,     8843648,      6412200,       733104,         3136;
  7: 128,   412400,   21582828,   154733440,    241771000,     86331312,      5051900,        8192;
  8: 256,  2073776,  191220624,  2354348864,   6769588000,   5297284944,   1041090064,    33180416,     20736;
  9: 512, 10395504, 1637082900, 32696793024, 157311623700, 226528738128, 100133928636, 11641478400, 210508956, 51200;

%F Formula

The row polynomials satisfy (9*n^2-6*n+1)*P(n,x) = (18*n^2+6*n-4-(216*n^3-216*n^2-252*n+208)*x+(729*n^4-1134*n^3+81*n^2+216*n-48)*x^2)*P(n-2,x) - ((108*n^2+36*n-69)*x-(216*n^3+324*n^2+117*n-218)*x^2-(324*n^4-1566*n^3+297*n^2+276*n-71)*x^3)*d/dx P(n-2,x) - ((81*n^2+27*n-45)*x^2-(324*n^3-27*n-30)*x^3+(324*n^3-81*n^2-54*n+15)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2-15*n+4+(108*n^3-108*n^2+36*n-4)*x)*P(n-1,x) + ((54*n^2-9*n-12)*x-(108*n^3-36*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 + 6*n - 4 - (216*n^3 - 216*n^2 - 252*n + 208)*x + (729*n^4 - 1134*n^3 + 81*n^2 + 216*n - 48)*x^2)*P[n - 2, x] - ((108*n^2 + 36*n - 69)*x - (216*n^3 + 324*n^2 + 117*n - 218)*x^2 - (324*n^4 - 1566*n^3 + 297*n^2 + 276*n - 71)*x^3)*D[P[n - 2, x], x] - ((81*n^2 + 27*n - 45)*x^2 - (324*n^3 - 27*n - 30)*x^3 + (324*n^3 - 81*n^2 - 54*n + 15)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 - 15*n + 4 + (108*n^3 - 108*n^2 + 36*n - 4)*x)*P[n - 1, x] + ((54*n^2 - 9*n - 12)*x - (108*n^3 - 36*n + 8)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 6*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^2 + 6*n - 4 - (216*n^3 - 216*n^2 - 252*n + 208)*x + (729*n^4 - 1134*n^3 + 81*n^2 + 216*n - 48)*x^2)*v[i - 2] - ((108*n^2 + 36*n - 69)*x - (216*n^3 + 324*n^2 + 117*n - 218)*x^2 - (324*n^4 - 1566*n^3 + 297*n^2 + 276*n - 71)*x^3)*deriv(v[i - 2], x) - ((81*n^2 + 27*n - 45)*x^2 - (324*n^3 - 27*n - 30)*x^3 + (324*n^3 - 81*n^2 - 54*n + 15)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 - 15*n + 4 + (108*n^3 - 108*n^2 + 36*n - 4)*x)*v[i - 1] + ((54*n^2 - 9*n - 12)*x - (108*n^3 - 36*n + 8)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 6*n + 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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.?
    
    ↑ Liu–Wang theorem (above)

References

[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.