A144699 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + n\right)\,P_{n}(x) = \left(-60\,x + 516\,x^{2} - 32\,n x + 409\,n x^{2} - 4\,n^{2} x + 106\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-10\,x + 101\,x^{2} - 20\,x^{3} - 2\,n x + 40\,n x^{2} + 67\,n x^{3} + 4\,n^{2} x^{2} + 34\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-5\,x^{2} + 81\,x^{3} - 76\,x^{4} - n x^{2} + 36\,n x^{3} - 35\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(4 + 64\,x + n + 32\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(9\,x - 80\,x^{2} + 2\,n x - 36\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 20\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	20							21
2	1	64	225						290
3	1	156	1539	2000					3696
4	1	344	7074	24416	15625				47460
5	1	724	27414	185696	307025	112500			633360
6	1	1488	96903	1132672	3566175	3327696	765625		8890560
7	1	3020	324189	6081136	31989275	55913220	32535559	5000000	131846400

All 15 rows

Technical details

Node id: n-26c0445816bc014a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-72b67609ab31c740

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,    20;
  2: 1,    64,     225;
  3: 1,   156,    1539,      2000;
  4: 1,   344,    7074,     24416,      15625;
  5: 1,   724,   27414,    185696,     307025,     112500;
  6: 1,  1488,   96903,   1132672,    3566175,    3327696,      765625;
  7: 1,  3020,  324189,   6081136,   31989275,   55913220,    32535559,    5000000;
  8: 1,  6088, 1047312,  30087904,  245462350,  703995768,   760477648,  294972544,   31640625;
  9: 1, 12228, 3306312, 140832384, 1697435250, 7405166664, 12989075232, 9312718848, 2524713381, 195312500;

%F Formula

The row polynomials satisfy (n+4)*P(n,x) = -((4*n^2+32*n+60)*x-(9*n^3+106*n^2+409*n+516)*x^2)*P(n-2,x) - ((2*n+10)*x-(4*n^2+40*n+101)*x^2-(4*n^3+34*n^2+67*n-20)*x^3)*d/dx P(n-2,x) - ((n+5)*x^2-(4*n^2+36*n+81)*x^3+(4*n^2+35*n+76)*x^4)*d^2/dx^2 P(n-2,x) + (n+4+(4*n^2+32*n+64)*x)*P(n-1,x) + ((2*n+9)*x-(4*n^2+36*n+80)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 20*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 20*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 32*n + 60)*x - (9*n^3 + 106*n^2 + 409*n + 516)*x^2)*P[n - 2, x] - ((2*n + 10)*x - (4*n^2 + 40*n + 101)*x^2 - (4*n^3 + 34*n^2 + 67*n - 20)*x^3)*D[P[n - 2, x], x] - ((n + 5)*x^2 - (4*n^2 + 36*n + 81)*x^3 + (4*n^2 + 35*n + 76)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 4 + (4*n^2 + 32*n + 64)*x)*P[n - 1, x] + ((2*n + 9)*x - (4*n^2 + 36*n + 80)*x^2)*D[P[n - 1, x], x])/(n + 4)]];
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 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^2 + 32*n + 60)*x - (9*n^3 + 106*n^2 + 409*n + 516)*x^2)*v[i - 2] - ((2*n + 10)*x - (4*n^2 + 40*n + 101)*x^2 - (4*n^3 + 34*n^2 + 67*n - 20)*x^3)*deriv(v[i - 2], x) - ((n + 5)*x^2 - (4*n^2 + 36*n + 81)*x^3 + (4*n^2 + 35*n + 76)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 4 + (4*n^2 + 32*n + 64)*x)*v[i - 1] + ((2*n + 9)*x - (4*n^2 + 36*n + 80)*x^2)*deriv(v[i - 1], x))/(n + 4)); 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 5-Eulerian numbers.?
  - 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.