A219512 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(12\,x + 42\,x^{2} + 24\,n x + 57\,n x^{2} - 36\,n^{2} x - 342\,n^{2} x^{2} + 243\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - x^{2} + 38\,x^{3} - 6\,n x + 75\,n x^{3} + 36\,n^{2} x^{2} - 270\,n^{2} x^{3} + 108\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 5\,x^{3} + 6\,x^{4} - 3\,n x^{2} - 12\,n x^{3} + 15\,n x^{4} + 36\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-2 + 3\,n - 24\,n x + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 8\,x^{2} + 6\,n x + 12\,n x^{2} - 36\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	12							13
2	1	48	135						184
3	1	132	1269	1680					3082
4	1	312	7074	29280	23625				60292
5	1	684	31050	280608	667125	374220			1353688
6	1	1440	119313	2005632	9912375	15694560	6621615		34354936
7	1	2964	422739	12052656	105970275	336822300	388143945	129729600	973144480

All 15 rows

Technical details

Node id: n-56859a54f6190f77 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-818310325c194a94

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: 1,    12;
  2: 1,    48,     135;
  3: 1,   132,    1269,      1680;
  4: 1,   312,    7074,     29280,      23625;
  5: 1,   684,   31050,    280608,     667125,      374220;
  6: 1,  1440,  119313,   2005632,    9912375,    15694560,      6621615;
  7: 1,  2964,  422739,  12052656,  105970275,   336822300,    388143945,    129729600;
  8: 1,  6024, 1421604,  64744416,  925329150,  5072877720,  11427606540,  10163180160,   2791213425;
  9: 1, 12156, 4617216, 322160064, 7054561350, 61084167768, 231897750840, 394748363520, 282376418625, 65472907500;

%F Formula

The row polynomials satisfy (3*n-2)*P(n,x) = -((36*n^2-24*n-12)*x-(243*n^3-342*n^2+57*n+42)*x^2)*P(n-2,x) - ((6*n+2)*x-(36*n^2-1)*x^2-(108*n^3-270*n^2+75*n+38)*x^3)*d/dx P(n-2,x) - ((3*n+1)*x^2-(36*n^2-12*n-5)*x^3+(36*n^2-15*n-6)*x^4)*d^2/dx^2 P(n-2,x) + (3*n-2+(36*n^2-24*n)*x)*P(n-1,x) + ((6*n-1)*x-(36*n^2-12*n-8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((36*n^2 - 24*n - 12)*x - (243*n^3 - 342*n^2 + 57*n + 42)*x^2)*P[n - 2, x] - ((6*n + 2)*x - (36*n^2 - 1)*x^2 - (108*n^3 - 270*n^2 + 75*n + 38)*x^3)*D[P[n - 2, x], x] - ((3*n + 1)*x^2 - (36*n^2 - 12*n - 5)*x^3 + (36*n^2 - 15*n - 6)*x^4)*D[P[n - 2, x], {x, 2}] + (3*n - 2 + (36*n^2 - 24*n)*x)*P[n - 1, x] + ((6*n - 1)*x - (36*n^2 - 12*n - 8)*x^2)*D[P[n - 1, x], x])/(3*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; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-((36*n^2 - 24*n - 12)*x - (243*n^3 - 342*n^2 + 57*n + 42)*x^2)*v[i - 2] - ((6*n + 2)*x - (36*n^2 - 1)*x^2 - (108*n^3 - 270*n^2 + 75*n + 38)*x^3)*deriv(v[i - 2], x) - ((3*n + 1)*x^2 - (36*n^2 - 12*n - 5)*x^3 + (36*n^2 - 15*n - 6)*x^4)*deriv(deriv(v[i - 2], x), x) + (3*n - 2 + (36*n^2 - 24*n)*x)*v[i - 1] + ((6*n - 1)*x - (36*n^2 - 12*n - 8)*x^2)*deriv(v[i - 1], x))/(3*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 third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.?
  - 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.