A156365 → θ+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1 + 16\,x + 12\,x^{2}\)

\(P_{1}(x) = 1 + 44\,x + 132\,x^{2} + 32\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	16	12															29
1	1	44	132	32														209
2	1	104	792	832	80													1809
3	1	228	3624	9664	4560	192												18269
4	1	480	14292	77312	95280	23040	448											210853
5	1	988	51516	499808	1249520	824256	110656	1024										2737769
6	1	2008	175296	2823488	12495200	16940928	6544384	514048	2304									39497657
7	1	4052	574080	14566144	104828320	251587968	203926016	48988160	2333952	5120								626813813
8	1	8144	1831644	70511616	779049120	3019055616	4362675072	2256371712	351675648	10424320	11264							10851614157
9	1	16332	5739252	326005920	5305477920	31202358912	72805504128	67910117376	23472426240	2448747520	45990912	24576						203522409089
10	1	32712	17756712	1457070400	33862522800	289079329536	1019277057792	1541756424192	975240656640	233131264000	16667633664	200982528	53248					4110690784225
11	1	65476	54447768	6349145152	205739374000	2465906469696	12531086889216	28642484318208	29590877636352	13167319936000	2234899093504	111509028864	871616512	114688				88957098135437
12	1	131008	165896868	27138909184	1203298361680	19733183794176	139501644088128	458279749091328	717437026738944	526218234511360	169424409324544	20842682253312	736139702272	3756261376	245760			2053407429309941
13	1	262076	503192940	114306728160	6830659083920	150166179513408	1436952463814336	6535986099686400	14705968724294400	16422313872163840	8809749198119936	2098378470580224	190206395658240	4809182658560	16101949440	524288		50361492158230169
14	1	524216	1521372240	476012782400	37868264084800	1097720058697344	13912932520877824	85132516541818880	264707437037299200	425662582709094400	349810874810642432	140508167513260032	25205116574842880	1706029812121600	31157703475200	68710039552	1114112	1307812949792047113

Technical details

Node id: n-81e7fb94c47d41ed · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f712c08c0fc317d2

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 2)
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,    16,      12;
  1: 1,    44,     132,        32;
  2: 1,   104,     792,       832,         80;
  3: 1,   228,    3624,      9664,       4560,         192;
  4: 1,   480,   14292,     77312,      95280,       23040,         448;
  5: 1,   988,   51516,    499808,    1249520,      824256,      110656,        1024;
  6: 1,  2008,  175296,   2823488,   12495200,    16940928,     6544384,      514048,        2304;
  7: 1,  4052,  574080,  14566144,  104828320,   251587968,   203926016,    48988160,     2333952,       5120;
  8: 1,  8144, 1831644,  70511616,  779049120,  3019055616,  4362675072,  2256371712,   351675648,   10424320,    11264;
  9: 1, 16332, 5739252, 326005920, 5305477920, 31202358912, 72805504128, 67910117376, 23472426240, 2448747520, 45990912, 24576;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2+16*n+12)*x^2*P(n-2,x) - ((2*n+6)*x^2-(4*n+12)*x^3)*d/dx P(n-2,x) + (1+(4*n+8)*x)*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 16*x + 12*x^2, P(1,x) = 1 + 44*x + 132*x^2 + 32*x^3.

%t Mathematica

P[0, x_] := 1 + 16*x + 12*x^2;
P[1, x_] := 1 + 44*x + 132*x^2 + 32*x^3;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 + 16*n + 12)*x^2*P[n - 2, x] - ((2*n + 6)*x^2 - (4*n + 12)*x^3)*D[P[n - 2, x], x] + (1 + (4*n + 8)*x)*P[n - 1, x] + (x - 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 16*x + 12*x^2; v[2] = 1 + 44*x + 132*x^2 + 32*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 + 16*n + 12)*x^2*v[i - 2] - ((2*n + 6)*x^2 - (4*n + 12)*x^3)*deriv(v[i - 2], x) + (1 + (4*n + 8)*x)*v[i - 1] + (x - 2*x^2)*deriv(v[i - 1], x)); 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 T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A142075 → θ+1

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.