A357367 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - n x + n^{2} x\right)\,P_{n}(x) = \left(2 - 2\,n - 7\,n x - 2\,n x^{2} - 2\,n^{2} x + n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 6\,x^{2} - 2\,x^{3} + 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	4														4
2	0	12	36													48
3	0	48	360	480												888
4	0	240	3240	10080	8400											21960
5	0	1440	30240	161280	302400	181440										676800
6	0	10080	302400	2419200	7560000	9979200	4656960									24927840
7	0	80640	3265920	36288000	166320000	359251200	363242880	138378240								1066826880
8	0	725760	38102400	558835200	3492720000	10897286400	17798901120	14529715200	4670265600							51986551680
9	0	7257600	479001600	8941363200	72648576000	305124019200	711956044800	929901772800	635156121600	176432256000						2840646412800
10	0	79833600	6466521600	149448499200	1525620096000	8238348518400	25630417612800	47424990412800	51447645849600	30169915776000	7374868300800					171967801420800
11	0	958003200	93405312000	2615348736000	32691859200000	219689293824000	871434198835200	2134124568576000	3258350903808000	3016991577600000	1548722343168000	337903056691200				11422617513753600
12	0	12454041600	1438441804800	47948060160000	719220902400000	5868842563584000	28757328561561600	89206406966476800	179209299709440000	232308351475200000	187395403523328000	85489473342873600	16838835658444800			825842561659315200
13	0	174356582400	23538138624000	920602755072000	16302340454400000	158458749216768000	936667273148006400	3568256278659072000	9032148705355776000	15332351197363200000	17240377124146176000	12310484161373798400	5051650697533440000	906706535454720000		64554347377955635200
14	0	2615348736000	407994402816000	18495746260992000	381474766632960000	4348812339615744000	30441686377310208000	139161994867703808000	430532421621958656000	916874601602319360000	1344749415683401728000	1333635784148828160000	853728967883151360000	318253993944606720000	52459449551308800000	5424587509142850048000

Technical details

Node id: n-560806510ad4c8da · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7844ae331c0f03a8

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: 0,       4;
  2: 0,      12,        36;
  3: 0,      48,       360,        480;
  4: 0,     240,      3240,      10080,        8400;
  5: 0,    1440,     30240,     161280,      302400,       181440;
  6: 0,   10080,    302400,    2419200,     7560000,      9979200,      4656960;
  7: 0,   80640,   3265920,   36288000,   166320000,    359251200,    363242880,    138378240;
  8: 0,  725760,  38102400,  558835200,  3492720000,  10897286400,  17798901120,  14529715200,   4670265600;
  9: 0, 7257600, 479001600, 8941363200, 72648576000, 305124019200, 711956044800, 929901772800, 635156121600, 176432256000;

%F Formula

The row polynomials satisfy -(2-(n^2-n)*x)*P(n,x) = -(2*n-2-(n^3-2*n^2-7*n)*x-(2*n^3-2*n)*x^2)*P(n-1,x) - (4*x-(2*n^2-6)*x^2-(2*n^2-2)*x^3)*d/dx 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[(-(2*n - 2 - (n^3 - 2*n^2 - 7*n)*x - (2*n^3 - 2*n)*x^2)*P[n - 1, x] - (4*x - (2*n^2 - 6)*x^2 - (2*n^2 - 2)*x^3)*D[P[n - 1, x], x])/(-2 + (n^2 - n)*x)]];
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] = (-(2*n - 2 - (n^3 - 2*n^2 - 7*n)*x - (2*n^3 - 2*n)*x^2)*v[i - 1] - (4*x - (2*n^2 - 6)*x^2 - (2*n^2 - 2)*x^3)*deriv(v[i - 1], x))/(-2 + (n^2 - n)*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

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) = binomial(n - 1, k - 1)*(n + k)! / k!.?
  - 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 for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A386789 → 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.