A290598 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2 - 2\,x + n\right)\,P_{n}(x) = \left(-6 + 4\,x - 2\,x^{2} + 7\,n - 5\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(-6\,x^{2} + 3\,n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 5 | 1 | 6 | ||||||
| 3 | 42 | 16 | 1 | 59 | |||||
| 4 | 490 | 270 | 33 | 1 | 794 | ||||
| 5 | 7280 | 5200 | 936 | 56 | 1 | 13473 | |||
| 6 | 131040 | 114400 | 27040 | 2400 | 85 | 1 | 274966 | ||
| 7 | 2766400 | 2845440 | 829920 | 97280 | 5130 | 120 | 1 | 6544291 | |
| 8 | 66946880 | 79119040 | 27387360 | 3979360 | 277970 | 9702 | 161 | 1 | 177720474 |
Compute more
?
Technical details
- Node id
n-aec89886b0fe4dfc· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-e38afe273a9e31c4
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: 5, 1; 3: 42, 16, 1; 4: 490, 270, 33, 1; 5: 7280, 5200, 936, 56, 1; 6: 131040, 114400, 27040, 2400, 85, 1; 7: 2766400, 2845440, 829920, 97280, 5130, 120, 1; 8: 66946880, 79119040, 27387360, 3979360, 277970, 9702, 161, 1; 9: 1825824000, 2434432000, 973772800, 168537600, 14630000, 677600, 16800, 208, 1; 10: 55383328000, 82162080000, 37246809600, 7471833600, 774219600, 44629200, 1470000, 27216, 261, 1;
%F FormulaThe row polynomials satisfy (n+2-2*x)*P(n,x) = (3*n^2+7*n-6-(5*n-4)*x-2*x^2)*P(n-1,x) + (3*n*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 + 7*n - 6 - (5*n - 4)*x - 2*x^2)*P[n - 1, x] + (3*n*x - 6*x^2)*D[P[n - 1, x], x])/(n + 2 - 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 + 7*n - 6 - (5*n - 4)*x - 2*x^2)*v[i - 1] + (3*n*x - 6*x^2)*deriv(v[i - 1], x))/(n + 2 - 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]
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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[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.