A156366 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-36\,x + 450\,x^{2} - 48\,n x + 837\,n x^{2} - 12\,n^{2} x + 468\,n^{2} x^{2} + 81\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 111\,x^{2} - 450\,x^{3} - 2\,n x + 72\,n x^{2} - 189\,n x^{3} + 12\,n^{2} x^{2} + 90\,n^{2} x^{3} + 36\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-3\,x^{2} + 75\,x^{3} - 198\,x^{4} - n x^{2} + 60\,n x^{3} - 171\,n x^{4} + 12\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(2 + 48\,x + n + 48\,n x + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(5\,x - 72\,x^{2} + 2\,n x - 60\,n x^{2} - 12\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 48\,x + 81\,x^{2}\)

\(P_{1}(x) = 1 + 132\,x + 891\,x^{2} + 432\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	48	81								130
1	1	132	891	432							1456
2	1	312	5346	11232	2025						18916
3	1	684	24462	130464	115425	8748					279784
4	1	1440	96471	1043712	2411775	1049760	35721				4638880
5	1	2964	347733	6747408	31628475	37555164	8823087	139968			85244800
6	1	6024	1183248	38117088	316284750	771871032	521812368	70263936	531441		1720069888
7	1	12156	3875040	196642944	2653466850	11462976792	16259913432	6696069120	538349733	1968300	37813274368

All 15 rows

Technical details

Node id: n-e190c0da2f2fe2af · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-a45daab6f850677d

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,    48,       81;
  1: 1,   132,      891,        432;
  2: 1,   312,     5346,      11232,         2025;
  3: 1,   684,    24462,     130464,       115425,          8748;
  4: 1,  1440,    96471,    1043712,      2411775,       1049760,         35721;
  5: 1,  2964,   347733,    6747408,     31628475,      37555164,       8823087,        139968;
  6: 1,  6024,  1183248,   38117088,    316284750,     771871032,     521812368,      70263936,        531441;
  7: 1, 12156,  3875040,  196642944,   2653466850,   11462976792,   16259913432,    6696069120,     538349733,      1968300;
  8: 1, 24432, 12363597,  951906816,  19719680850,  137555721504,  347855170194,  308417808384,   81117559917,   4007458800,     7144929;
  9: 1, 48996, 38739951, 4401079920, 134294909850, 1421657477928, 5805101368206, 9282464168832, 5414153504085, 941380809300, 29172745107, 25509168;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -((12*n^2+48*n+36)*x-(81*n^3+468*n^2+837*n+450)*x^2)*P(n-2,x) - ((2*n+6)*x-(12*n^2+72*n+111)*x^2-(36*n^3+90*n^2-189*n-450)*x^3)*d/dx P(n-2,x) - ((n+3)*x^2-(12*n^2+60*n+75)*x^3+(36*n^2+171*n+198)*x^4)*d^2/dx^2 P(n-2,x) + (n+2+(12*n^2+48*n+48)*x)*P(n-1,x) + ((2*n+5)*x-(12*n^2+60*n+72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 48*x + 81*x^2, P(1,x) = 1 + 132*x + 891*x^2 + 432*x^3.

%t Mathematica

P[0, x_] := 1 + 48*x + 81*x^2;
P[1, x_] := 1 + 132*x + 891*x^2 + 432*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((12*n^2 + 48*n + 36)*x - (81*n^3 + 468*n^2 + 837*n + 450)*x^2)*P[n - 2, x] - ((2*n + 6)*x - (12*n^2 + 72*n + 111)*x^2 - (36*n^3 + 90*n^2 - 189*n - 450)*x^3)*D[P[n - 2, x], x] - ((n + 3)*x^2 - (12*n^2 + 60*n + 75)*x^3 + (36*n^2 + 171*n + 198)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 2 + (12*n^2 + 48*n + 48)*x)*P[n - 1, x] + ((2*n + 5)*x - (12*n^2 + 60*n + 72)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 48*x + 81*x^2; v[2] = 1 + 132*x + 891*x^2 + 432*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-((12*n^2 + 48*n + 36)*x - (81*n^3 + 468*n^2 + 837*n + 450)*x^2)*v[i - 2] - ((2*n + 6)*x - (12*n^2 + 72*n + 111)*x^2 - (36*n^3 + 90*n^2 - 189*n - 450)*x^3)*deriv(v[i - 2], x) - ((n + 3)*x^2 - (12*n^2 + 60*n + 75)*x^3 + (36*n^2 + 171*n + 198)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 2 + (12*n^2 + 48*n + 48)*x)*v[i - 1] + ((2*n + 5)*x - (12*n^2 + 60*n + 72)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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 of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.