basis-map coefficients of A021012 → basis A008277 → (θ+1)²
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-8\,x + 18\,x^{2} + 12\,n x - 27\,n x^{2} - 4\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 8\,x^{2} + 8\,x^{3} + 6\,n x - 12\,n x^{2} - 12\,n x^{3} - 2\,n^{2} x + 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 8\,x^{3} + 3\,n x^{2} - 12\,n x^{3} - n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(1 + 4\,x - n - 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} - 2\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -1 - 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | -1 | -4 | -5 | ||||||
| 3 | 2 | 24 | 18 | 44 | |||||
| 4 | -6 | -168 | -324 | -96 | -594 | ||||
| 5 | 24 | 1440 | 5400 | 3840 | 600 | 11304 | |||
| 6 | -120 | -14880 | -97200 | -124800 | -45000 | -4320 | -286320 | ||
| 7 | 720 | 181440 | 1950480 | 4032000 | 2520000 | 544320 | 35280 | 9264240 | |
| 8 | -5040 | -2560320 | -43817760 | -137168640 | -132300000 | -48263040 | -6914880 | -322560 | -371352240 |
Compute more
?
Technical details
- Node id
n-1e68a1963f2d6543· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-915ca4b22aed877c
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 -1)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): no - 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: -1, -4; 3: 2, 24, 18; 4: -6, -168, -324, -96; 5: 24, 1440, 5400, 3840, 600; 6: -120, -14880, -97200, -124800, -45000, -4320; 7: 720, 181440, 1950480, 4032000, 2520000, 544320, 35280; 8: -5040, -2560320, -43817760, -137168640, -132300000, -48263040, -6914880, -322560; 9: 40320, 41126400, 1097712000, 5012582400, 7006608000, 3840721920, 912764160, 92897280, 3265920; 10: -362880, -741726720, -30471033600, -198016358400, -385786800000, -298204623360, -104552985600, -17418240000, -1322697600, -36288000;
%F FormulaThe row polynomials satisfy P(n,x) = -((4*n^2-12*n+8)*x-(9*n^2-27*n+18)*x^2)*P(n-2,x) - ((2*n^2-6*n+4)*x-(4*n^2-12*n+8)*x^2-(4*n^2-12*n+8)*x^3)*d/dx P(n-2,x) - ((n^2-3*n+2)*x^2-(4*n^2-12*n+8)*x^3)*d^2/dx^2 P(n-2,x) - (n-1+(4*n-4)*x)*P(n-1,x) - ((2*n-2)*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 - 4*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := -1 - 4*x;
P[n_, x_] := P[n, x] = Expand[-((4*n^2 - 12*n + 8)*x - (9*n^2 - 27*n + 18)*x^2)*P[n - 2, x] - ((2*n^2 - 6*n + 4)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], x] - ((n^2 - 3*n + 2)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], {x, 2}] - (n - 1 + (4*n - 4)*x)*P[n - 1, x] - ((2*n - 2)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 - 4*x; for(n = 3, N, my(i = n); v[i] = -((4*n^2 - 12*n + 8)*x - (9*n^2 - 27*n + 18)*x^2)*v[i - 2] - ((2*n^2 - 6*n + 4)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(v[i - 2], x) - ((n^2 - 3*n + 2)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(deriv(v[i - 2], x), x) - (n - 1 + (4*n - 4)*x)*v[i - 1] - ((2*n - 2)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from 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
basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
-
Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
-
from
basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
-
Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
-
-
from
basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
Related ?
Same fingerprint ?
- A048993 → D+1 → (θ+1)²
- A008277 → θ+1 → θ+1
- basis-map coefficients of A112227 → basis A008277 → (θ+1)²
- A008277 → (θ+1)²
- A008277 → (θ+1)² → reverse
- A130123 → basis A008277 → (θ+1)²
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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.