A048854 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-3\,x + n\right)\,P_{n}(x) = \left(-6 + 15\,x - 3\,x^{2} - 2\,n - 11\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(-12\,x - 12\,x^{2} + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 3 | 1 | 4 | ||||||
| 3 | 24 | 14 | 1 | 39 | |||||
| 4 | 300 | 240 | 33 | 1 | 574 | ||||
| 5 | 5040 | 5040 | 1008 | 60 | 1 | 11149 | |||
| 6 | 105840 | 126000 | 32760 | 2880 | 95 | 1 | 267576 | ||
| 7 | 2661120 | 3659040 | 1164240 | 134640 | 6600 | 138 | 1 | 7625779 | |
| 8 | 77837760 | 121080960 | 45405360 | 6486480 | 420420 | 13104 | 189 | 1 | 251244274 |
Compute more
?
Technical details
- Node id
n-92baaccdeb9edfec· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-aa579d4e84ab5b11
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: 24, 14, 1; 4: 300, 240, 33, 1; 5: 5040, 5040, 1008, 60, 1; 6: 105840, 126000, 32760, 2880, 95, 1; 7: 2661120, 3659040, 1164240, 134640, 6600, 138, 1; 8: 77837760, 121080960, 45405360, 6486480, 420420, 13104, 189, 1; 9: 2594592000, 4497292800, 1937295360, 328648320, 26426400, 1092000, 23520, 248, 1; 10: 97037740800, 185253868800, 89980450560, 17643225600, 1690809120, 86879520, 2484720, 39168, 315, 1;
%F FormulaThe row polynomials satisfy (n-3*x)*P(n,x) = (4*n^2-2*n-6-(11*n-15)*x-3*x^2)*P(n-1,x) + ((4*n-12)*x-12*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[((4*n^2 - 2*n - 6 - (11*n - 15)*x - 3*x^2)*P[n - 1, x] + ((4*n - 12)*x - 12*x^2)*D[P[n - 1, x], x])/(n - 3*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] = ((4*n^2 - 2*n - 6 - (11*n - 15)*x - 3*x^2)*v[i - 1] + ((4*n - 12)*x - 12*x^2)*deriv(v[i - 1], x))/(n - 3*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 differential operators
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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
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].?
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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[4,1].?
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.