A035342 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(8\,n - 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(12\,x - 12\,x^{2} + 4\,x^{3} + 8\,n + 14\,n x - 12\,n x^{2} + 8\,n x^{3} + 16\,n^{2} - 12\,n^{2} x + 4\,n^{2} x^{2} + 4\,n^{2} x^{3} + 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(9\,x - 13\,x^{2} + 6\,x^{3} + 8\,n x - 12\,n x^{2} + 12\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-1}'(x) + \left(2\,x - 3\,x^{2} + 2\,x^{3} + 4\,n x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 4 | 7 | ||||||
| 2 | 15 | 36 | 9 | 60 | |||||
| 3 | 105 | 348 | 162 | 16 | 631 | ||||
| 4 | 945 | 3900 | 2565 | 480 | 25 | 7915 | |||
| 5 | 10395 | 50580 | 42120 | 11280 | 1125 | 36 | 115536 | ||
| 6 | 135135 | 749700 | 745605 | 255360 | 36750 | 2268 | 49 | 1924867 | |
| 7 | 2027025 | 12535740 | 14362110 | 5922000 | 1097250 | 98280 | 4116 | 64 | 36046585 |
Compute more
?
Technical details
- Node id
n-34a6ffa52e7a8735· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-11e33413962f9e56
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: 3, 4; 2: 15, 36, 9; 3: 105, 348, 162, 16; 4: 945, 3900, 2565, 480, 25; 5: 10395, 50580, 42120, 11280, 1125, 36; 6: 135135, 749700, 745605, 255360, 36750, 2268, 49; 7: 2027025, 12535740, 14362110, 5922000, 1097250, 98280, 4116, 64; 8: 34459425, 233751420, 301085505, 143972640, 32295375, 3742200, 228438, 6912, 81; 9: 654729075, 4812866100, 6848736300, 3702661200, 967750875, 136318140, 10804500, 478080, 10935, 100;
%F FormulaThe row polynomials satisfy (8*n-6*n*x+4*n^2*x^2)*P(n,x) = (16*n^2+8*n-(12*n^2-14*n-12)*x+(8*n^3+4*n^2-12*n-12)*x^2+(4*n^2+8*n+4)*x^3)*P(n-1,x) + ((8*n+9)*x-(12*n+13)*x^2+(4*n^2+12*n+6)*x^3)*d/dx P(n-1,x) + (2*x-3*x^2+(4*n+2)*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 + 8*n - (12*n^2 - 14*n - 12)*x + (8*n^3 + 4*n^2 - 12*n - 12)*x^2 + (4*n^2 + 8*n + 4)*x^3)*P[n - 1, x] + ((8*n + 9)*x - (12*n + 13)*x^2 + (4*n^2 + 12*n + 6)*x^3)*D[P[n - 1, x], x] + (2*x - 3*x^2 + (4*n + 2)*x^3)*D[P[n - 1, x], {x, 2}])/(8*n - 6*n*x + 4*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^2 + 8*n - (12*n^2 - 14*n - 12)*x + (8*n^3 + 4*n^2 - 12*n - 12)*x^2 + (4*n^2 + 8*n + 4)*x^3)*v[i - 1] + ((8*n + 9)*x - (12*n + 13)*x^2 + (4*n^2 + 12*n + 6)*x^3)*deriv(v[i - 1], x) + (2*x - 3*x^2 + (4*n + 2)*x^3)*deriv(deriv(v[i - 1], x), x))/(8*n - 6*n*x + 4*n^2*x^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 ›
-
from
The convolution matrix of the double factorial of odd numbers (A001147).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
-
Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
-
from
Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Parent-row provers of polytool (above)
-
from
Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
-
Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
-
-
from
basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
-
Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
-
from
The convolution matrix of the double factorial of odd numbers (A001147).?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [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
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.