A358735 → basis A007318 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-2\,n + 8\,n x - 2\,n^{2} - 12\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-3}(x) + \left(-8\,n x^{2} + 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-3}'(x) + \left(-2 + 6\,n - 8\,n x - 18\,n x^{2} + n^{2} + 4\,n^{2} x + 45\,n^{2} x^{2} + 8\,n^{3} x - 36\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(8\,n x^{2} - 8\,n x^{3} - 4\,n^{2} x^{2} + 20\,n^{2} x^{3} - 8\,n^{3} x^{2} - 16\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4 - 6\,n + 8\,n x - 2\,n^{2} - 12\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-8\,n x^{2} + 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)
\(P_{2}(x) = 7 + 32\,x + 18\,x^{2}\)
\(P_{3}(x) = 33 + 240\,x + 306\,x^{2} + 96\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 4 | 6 | ||||||
| 2 | 7 | 32 | 18 | 57 | |||||
| 3 | 33 | 240 | 306 | 96 | 675 | ||||
| 4 | 191 | 1936 | 3978 | 2752 | 600 | 9457 | |||
| 5 | 1304 | 17132 | 49320 | 54528 | 25700 | 4320 | 152304 | ||
| 6 | 10241 | 166416 | 621864 | 959552 | 716200 | 256608 | 35280 | 2766161 | |
| 7 | 90865 | 1767360 | 8168508 | 16320192 | 16915200 | 9524448 | 2762424 | 322560 | 55871557 |
Compute more
?
Technical details
- Node id
n-a13ac7f16521f585· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-0f8931d9867a48c4
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: 2, 4; 2: 7, 32, 18; 3: 33, 240, 306, 96; 4: 191, 1936, 3978, 2752, 600; 5: 1304, 17132, 49320, 54528, 25700, 4320; 6: 10241, 166416, 621864, 959552, 716200, 256608, 35280; 7: 90865, 1767360, 8168508, 16320192, 16915200, 9524448, 2762424, 322560; 8: 898409, 20414864, 112875696, 278416704, 372438200, 289850976, 131299616, 32090112, 3265920; 9: 9791634, 255138868, 1646865144, 4852274688, 7995139900, 8005635360, 4992207808, 1896109056, 401451552, 36288000;
%F FormulaThe row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (2*n^3-2*n^2-2*n+(4*n^3-12*n^2+8*n)*x)*P(n-3,x) - (4*n^3-12*n^2+8*n)*x^2*d/dx P(n-3,x) - (n^4-n^2-6*n+2+(4*n^4-8*n^3-4*n^2+8*n)*x-(9*n^4-36*n^3+45*n^2-18*n)*x^2)*P(n-2,x) + ((4*n^4-8*n^3-4*n^2+8*n)*x^2+(4*n^4-16*n^3+20*n^2-8*n)*x^3)*d/dx P(n-2,x) + (2*n^3-2*n^2-6*n+4+(4*n^3-12*n^2+8*n)*x)*P(n-1,x) - (4*n^3-12*n^2+8*n)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 7 + 32*x + 18*x^2, P(3,x) = 33 + 240*x + 306*x^2 + 96*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 7 + 32*x + 18*x^2;
P[3, x_] := 33 + 240*x + 306*x^2 + 96*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (2*n^3 - 2*n^2 - 2*n + (4*n^3 - 12*n^2 + 8*n)*x)*P[n - 3, x] - (4*n^3 - 12*n^2 + 8*n)*x^2*D[P[n - 3, x], x] - (n^4 - n^2 - 6*n + 2 + (4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x - (9*n^4 - 36*n^3 + 45*n^2 - 18*n)*x^2)*P[n - 2, x] + ((4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x^2 + (4*n^4 - 16*n^3 + 20*n^2 - 8*n)*x^3)*D[P[n - 2, x], x] + (2*n^3 - 2*n^2 - 6*n + 4 + (4*n^3 - 12*n^2 + 8*n)*x)*P[n - 1, x] - (4*n^3 - 12*n^2 + 8*n)*x^2*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 7 + 32*x + 18*x^2; v[4] = 33 + 240*x + 306*x^2 + 96*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (2*n^3 - 2*n^2 - 2*n + (4*n^3 - 12*n^2 + 8*n)*x)*v[i - 3] - (4*n^3 - 12*n^2 + 8*n)*x^2*deriv(v[i - 3], x) - (n^4 - n^2 - 6*n + 2 + (4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x - (9*n^4 - 36*n^3 + 45*n^2 - 18*n)*x^2)*v[i - 2] + ((4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x^2 + (4*n^4 - 16*n^3 + 20*n^2 - 8*n)*x^3)*deriv(v[i - 2], x) + (2*n^3 - 2*n^2 - 6*n + 4 + (4*n^3 - 12*n^2 + 8*n)*x)*v[i - 1] - (4*n^3 - 12*n^2 + 8*n)*x^2*deriv(v[i - 1], x))/(n^2 - 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]
-
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
Proof chains ? JSON
-
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
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023