A257625 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(9 - 54\,x - 27\,x^{2} + 18\,n - 72\,n x + 18\,n x^{2} + 72\,n^{2} x + 108\,n^{2} x^{2} - 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(18\,x - 18\,x^{3} + 36\,n x - 36\,n x^{2} - 72\,n^{2} x^{2} + 72\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-6 + 6\,x - 24\,n x + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 12\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	9	108	27													144
3	27	1242	1863	108												3240
4	81	12312	55890	24624	405											93312
5	243	115182	1226178	1634904	287955	1458										3265920
6	729	1052676	23057541	68665968	38429235	3158028	5103									134369280
7	2187	9530946	397970577	2272056804	2840071005	795941154	33358311	17496								6348948480
8	6561	85975344	6527118996	65223898272	153363834270	97835847408	15229944324	343901376	59049							338610585600
9	19683	774447318	103710120660	1709112051504	6831309329010	8197571194812	2990946090132	276560321760	3485012931	196830						20113468784640
10	59049	6972269724	1614897029727	42113960321184	267764564394090	541039698857592	374870390151726	84227920642368	4844691089181	34861348620	649539					1316517956812800
11	177147	62757867690	24819584507307	994206164766300	9598553506707030	30385691675943948	35449973621934606	15357685610731248	2236963870724175	82731948357690	345168272295	2125764				94131033912115200
12	531441	564845255496	378224386106310	22764617882647632	322541587684941435	1521049714668329904	2765097942510899268	2028066286224439872	580574857832894583	56911544706619080	1386822749056470	3389071532976	6908733			7298775552570163200
13	1594323	5083687015614	5731827275538486	509827823968530312	10330601891632062525	69934185750373302654	187693929778012391412	214507348317728447328	104901278625559953981	20661203783264125050	1402026515913458358	22927309102153944	33043965601491	22320522		609969099750506496000
14	4782969	45753441420852	86549323922333865	11233712412688677264	319075192211587450245	3014441299149952064292	11481147313489694772981	19305661348595560259520	14761475117343893279547	5024068831916586773820	701965422865492390539	33701137238066031792	375047070330113415	320274089945964	71744535	54653231337645382041600

Technical details

Node id: n-cb35305b0e1e69ed · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2e37e4aadfbd49d5

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:     3,         6;
  2:     9,       108,           27;
  3:    27,      1242,         1863,           108;
  4:    81,     12312,        55890,         24624,           405;
  5:   243,    115182,      1226178,       1634904,        287955,          1458;
  6:   729,   1052676,     23057541,      68665968,      38429235,       3158028,          5103;
  7:  2187,   9530946,    397970577,    2272056804,    2840071005,     795941154,      33358311,        17496;
  8:  6561,  85975344,   6527118996,   65223898272,  153363834270,   97835847408,   15229944324,    343901376,      59049;
  9: 19683, 774447318, 103710120660, 1709112051504, 6831309329010, 8197571194812, 2990946090132, 276560321760, 3485012931, 196830;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = (18*n+9+(72*n^2-72*n-54)*x-(72*n^3-108*n^2-18*n+27)*x^2)*P(n-2,x) + ((36*n+18)*x-(72*n^2+36*n)*x^2+(72*n^2-18)*x^3)*d/dx P(n-2,x) - (6-(24*n^2-24*n+6)*x)*P(n-1,x) + ((12*n-6)*x-(12*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 9 + (72*n^2 - 72*n - 54)*x - (72*n^3 - 108*n^2 - 18*n + 27)*x^2)*P[n - 2, x] + ((36*n + 18)*x - (72*n^2 + 36*n)*x^2 + (72*n^2 - 18)*x^3)*D[P[n - 2, x], x] - (6 - (24*n^2 - 24*n + 6)*x)*P[n - 1, x] + ((12*n - 6)*x - (12*n - 6)*x^2)*D[P[n - 1, x], x])/(2*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] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((18*n + 9 + (72*n^2 - 72*n - 54)*x - (72*n^3 - 108*n^2 - 18*n + 27)*x^2)*v[i - 2] + ((36*n + 18)*x - (72*n^2 + 36*n)*x^2 + (72*n^2 - 18)*x^3)*deriv(v[i - 2], x) - (6 - (24*n^2 - 24*n + 6)*x)*v[i - 1] + ((12*n - 6)*x - (12*n - 6)*x^2)*deriv(v[i - 1], x))/(2*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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - 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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.