A290598 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2 - x + n x\right)\,P_{n}(x) = \left(4 - 2\,x - 6\,n - 5\,n x + n x^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 4 | 2 | 6 | ||||||
| 3 | 28 | 28 | 3 | 59 | |||||
| 4 | 280 | 420 | 90 | 4 | 794 | ||||
| 5 | 3640 | 7280 | 2340 | 208 | 5 | 13473 | |||
| 6 | 58240 | 145600 | 62400 | 8320 | 400 | 6 | 274966 | ||
| 7 | 1106560 | 3319680 | 1778400 | 316160 | 22800 | 684 | 7 | 6544291 | |
| 8 | 24344320 | 85205120 | 54774720 | 12172160 | 1170400 | 52668 | 1078 | 8 | 177720474 |
Compute more
?
Technical details
- Node id
n-ed7f37079a43bdf8· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-9c7ee164876013bf
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: 4, 2; 3: 28, 28, 3; 4: 280, 420, 90, 4; 5: 3640, 7280, 2340, 208, 5; 6: 58240, 145600, 62400, 8320, 400, 6; 7: 1106560, 3319680, 1778400, 316160, 22800, 684, 7; 8: 24344320, 85205120, 54774720, 12172160, 1170400, 52668, 1078, 8; 9: 608608000, 2434432000, 1825824000, 486886400, 58520000, 3511200, 107800, 1600, 9; 10: 17041024000, 76684608000, 65729664000, 20449228800, 2949408000, 221205600, 9055200, 201600, 2268, 10;
%F FormulaThe row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(6*n-4-(3*n^2-5*n-2)*x-n*x^2)*P(n-1,x) - (6*x-3*n*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[(-(6*n - 4 - (3*n^2 - 5*n - 2)*x - n*x^2)*P[n - 1, x] - (6*x - 3*n*x^2)*D[P[n - 1, x], x])/(-2 + (n - 1)*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] = (-(6*n - 4 - (3*n^2 - 5*n - 2)*x - n*x^2)*v[i - 1] - (6*x - 3*n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 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]
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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
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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
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,2].?
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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
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from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,2].?
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.