A372873 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(4 + 2\,n + 16\,n x - 4\,n^{2} - 48\,n^{2} x - 2\,n^{3} + 32\,n^{3} x\right)\,P_{n}(x) = \left(-2 + 48\,x + 96\,x^{2} - n + 20\,n x - 256\,n x^{2} + 2\,n^{2} - 40\,n^{2} x - 32\,n^{2} x^{2} + n^{3} - 28\,n^{3} x + 192\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x + 64\,x^{3} - 6\,n x + 56\,n x^{2} - 128\,n x^{3} - 2\,n^{2} x + 40\,n^{2} x^{2} - 128\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 0 | 4 | 4 | ||||||
| 2 | 0 | 2 | 12 | 14 | |||||
| 3 | 0 | 0 | 18 | 32 | 50 | ||||
| 4 | 0 | 0 | 3 | 96 | 80 | 179 | |||
| 5 | 0 | 0 | 0 | 40 | 400 | 192 | 632 | ||
| 6 | 0 | 0 | 0 | 4 | 300 | 1440 | 448 | 2192 | |
| 7 | 0 | 0 | 0 | 0 | 70 | 1680 | 4704 | 1024 | 7478 |
Compute more
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Technical details
- Node id
n-8107172ab43297c1· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-1793c0ef809343ce
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 ExampleTriangle begins: 0: 1; 1: 0, 4; 2: 0, 2, 12; 3: 0, 0, 18, 32; 4: 0, 0, 3, 96, 80; 5: 0, 0, 0, 40, 400, 192; 6: 0, 0, 0, 4, 300, 1440, 448; 7: 0, 0, 0, 0, 70, 1680, 4704, 1024; 8: 0, 0, 0, 0, 5, 672, 7840, 14336, 2304; 9: 0, 0, 0, 0, 0, 108, 4704, 32256, 41472, 5120;
%F FormulaThe row polynomials satisfy -(2*n^3+4*n^2-2*n-4-(32*n^3-48*n^2+16*n)*x)*P(n,x) = (n^3+2*n^2-n-2-(28*n^3+40*n^2-20*n-48)*x+(192*n^3-32*n^2-256*n+96)*x^2)*P(n-1,x) - ((2*n^2+6*n+4)*x-(40*n^2+56*n)*x^2+(128*n^2+128*n-64)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 2*n^2 - n - 2 - (28*n^3 + 40*n^2 - 20*n - 48)*x + (192*n^3 - 32*n^2 - 256*n + 96)*x^2)*P[n - 1, x] - ((2*n^2 + 6*n + 4)*x - (40*n^2 + 56*n)*x^2 + (128*n^2 + 128*n - 64)*x^3)*D[P[n - 1, x], x])/(-2*n^3 - 4*n^2 + 2*n + 4 + (32*n^3 - 48*n^2 + 16*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 2*n^2 - n - 2 - (28*n^3 + 40*n^2 - 20*n - 48)*x + (192*n^3 - 32*n^2 - 256*n + 96)*x^2)*v[i - 1] - ((2*n^2 + 6*n + 4)*x - (40*n^2 + 56*n)*x^2 + (128*n^2 + 128*n - 64)*x^3)*deriv(v[i - 1], x))/(-2*n^3 - 4*n^2 + 2*n + 4 + (32*n^3 - 48*n^2 + 16*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]
-
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
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.