A185410 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(n - 8\,n x - 8\,n x^{2} + 4\,n^{2} x + 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,n x + 2\,n x^{2} - 4\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 + 4\,x - 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 1 | 4 | 5 | |||||
| 3 | 1 | 20 | 12 | 33 | ||||
| 4 | 1 | 72 | 180 | 32 | 285 | |||
| 5 | 1 | 232 | 1548 | 1184 | 80 | 3045 | ||
| 6 | 1 | 716 | 10524 | 20672 | 6640 | 192 | 38745 | |
| 7 | 1 | 2172 | 63360 | 256960 | 214800 | 33984 | 448 | 571725 |
Compute more
?
Technical details
- Node id
n-8854f7202c6a83aa· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-6c0433a6238a5048
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 1; 2: 1, 4; 3: 1, 20, 12; 4: 1, 72, 180, 32; 5: 1, 232, 1548, 1184, 80; 6: 1, 716, 10524, 20672, 6640, 192; 7: 1, 2172, 63360, 256960, 214800, 33984, 448; 8: 1, 6544, 355896, 2643520, 4502800, 1920384, 164416, 1024; 9: 1, 19664, 1916904, 24198976, 73569200, 64351104, 15580096, 766976, 2304;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (n+(4*n^2-8*n)*x-(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) + (2*n*x-(4*n^2-2*n)*x^2+(4*n^2-4*n)*x^3)*d/dx P(n-2,x) - (1-(4*n^2-8*n+4)*x)*P(n-1,x) + ((2*n-2)*x-(2*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (4*n^2 - 8*n)*x - (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] + (2*n*x - (4*n^2 - 2*n)*x^2 + (4*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] - (1 - (4*n^2 - 8*n + 4)*x)*P[n - 1, x] + ((2*n - 2)*x - (2*n - 2)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + (4*n^2 - 8*n)*x - (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] + (2*n*x - (4*n^2 - 2*n)*x^2 + (4*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) - (1 - (4*n^2 - 8*n + 4)*x)*v[i - 1] + ((2*n - 2)*x - (2*n - 2)*x^2)*deriv(v[i - 1], x))/(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]
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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
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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
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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.