A176230 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-3 + n x\right)\,P_{n}(x) = \left(3 - 6\,x + x^{2} - 6\,n - n x + n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 2\,x^{2} + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 3 | 12 | 3 | 18 | |||||
| 3 | 15 | 90 | 45 | 4 | 154 | ||||
| 4 | 105 | 840 | 630 | 112 | 5 | 1692 | |||
| 5 | 945 | 9450 | 9450 | 2520 | 225 | 6 | 22596 | ||
| 6 | 10395 | 124740 | 155925 | 55440 | 7425 | 396 | 7 | 354328 | |
| 7 | 135135 | 1891890 | 2837835 | 1261260 | 225225 | 18018 | 637 | 8 | 6370008 |
Compute more
?
Technical details
- Node id
n-5a42344e07f62baa· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-6ab721de33bab167
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: 1, 2; 2: 3, 12, 3; 3: 15, 90, 45, 4; 4: 105, 840, 630, 112, 5; 5: 945, 9450, 9450, 2520, 225, 6; 6: 10395, 124740, 155925, 55440, 7425, 396, 7; 7: 135135, 1891890, 2837835, 1261260, 225225, 18018, 637, 8; 8: 2027025, 32432400, 56756700, 30270240, 6756750, 720720, 38220, 960, 9; 9: 34459425, 620269650, 1240539300, 771891120, 206756550, 27567540, 1949220, 73440, 1377, 10;
%F FormulaThe row polynomials satisfy -(3-n*x)*P(n,x) = -(6*n-3-(2*n^2-n-6)*x-(n+1)*x^2)*P(n-1,x) - (6*x-(2*n+2)*x^2)*d/dx 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[(-(6*n - 3 - (2*n^2 - n - 6)*x - (n + 1)*x^2)*P[n - 1, x] - (6*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(-3 + n*x)]];
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] = (-(6*n - 3 - (2*n^2 - n - 6)*x - (n + 1)*x^2)*v[i - 1] - (6*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(-3 + n*x)); 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
Proof chains ? JSON
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
from
Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].?
-
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
Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].?
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.