A290596 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2\,x + n\right)\,P_{n}(x) = \left(-4 + 6\,x - 2\,x^{2} - n - 5\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(-6\,x - 6\,x^{2} + 3\,n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 3 | 1 | 4 | ||||||
| 3 | 20 | 12 | 1 | 33 | |||||
| 4 | 200 | 168 | 27 | 1 | 396 | ||||
| 5 | 2640 | 2816 | 660 | 48 | 1 | 6165 | |||
| 6 | 43120 | 55440 | 16940 | 1820 | 75 | 1 | 117396 | ||
| 7 | 837760 | 1256640 | 471240 | 66640 | 4080 | 108 | 1 | 2636469 | |
| 8 | 18849600 | 32253760 | 14294280 | 2499000 | 202300 | 7980 | 147 | 1 | 68107068 |
Compute more
?
Technical details
- Node id
n-6d54ebf30881a6d6· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-3e125a4b64c05092
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): 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 3, 1; 3: 20, 12, 1; 4: 200, 168, 27, 1; 5: 2640, 2816, 660, 48, 1; 6: 43120, 55440, 16940, 1820, 75, 1; 7: 837760, 1256640, 471240, 66640, 4080, 108, 1; 8: 18849600, 32253760, 14294280, 2499000, 202300, 7980, 147, 1; 9: 481712000, 924887040, 472077760, 98094080, 9853200, 515200, 14168, 192, 1; 10: 13776963200, 29307358080, 16908091200, 4064773440, 486748080, 31646160, 1155336, 23400, 243, 1;
%F FormulaThe row polynomials satisfy (n-2*x)*P(n,x) = (3*n^2-n-4-(5*n-6)*x-2*x^2)*P(n-1,x) + ((3*n-6)*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - n - 4 - (5*n - 6)*x - 2*x^2)*P[n - 1, x] + ((3*n - 6)*x - 6*x^2)*D[P[n - 1, x], x])/(n - 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n^2 - n - 4 - (5*n - 6)*x - 2*x^2)*v[i - 1] + ((3*n - 6)*x - 6*x^2)*deriv(v[i - 1], x))/(n - 2*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 differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].?
-
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
-
from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].?
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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.