A286724 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 - 6\,x + 16\,x^{2} + 4\,n x - 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 20\,x^{3} - 8\,n x^{3}\right)\,P_{n-1}'(x) + \left(4\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 2 | 3 | ||||||
| 3 | 1 | 8 | 8 | 17 | |||||
| 4 | 1 | 18 | 72 | 48 | 139 | ||||
| 5 | 1 | 32 | 288 | 768 | 384 | 1473 | |||
| 6 | 1 | 50 | 800 | 4800 | 9600 | 3840 | 19091 | ||
| 7 | 1 | 72 | 1800 | 19200 | 86400 | 138240 | 46080 | 291793 | |
| 8 | 1 | 98 | 3528 | 58800 | 470400 | 1693440 | 2257920 | 645120 | 5129307 |
Compute more
?
Technical details
- Node id
n-17e55966d5083866· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-31dfc9d837f82561
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: 1, 2; 3: 1, 8, 8; 4: 1, 18, 72, 48; 5: 1, 32, 288, 768, 384; 6: 1, 50, 800, 4800, 9600, 3840; 7: 1, 72, 1800, 19200, 86400, 138240, 46080; 8: 1, 98, 3528, 58800, 470400, 1693440, 2257920, 645120; 9: 1, 128, 6272, 150528, 1881600, 12042240, 36126720, 41287680, 10321920; 10: 1, 162, 10368, 338688, 6096384, 60963840, 325140480, 836075520, 836075520, 185794560;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(4*n-6)*x+(4*n^2-16*n+16)*x^2)*P(n-1,x) - (4*x^2+(8*n-20)*x^3)*d/dx P(n-1,x) + 4*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 6)*x + (4*n^2 - 16*n + 16)*x^2)*P[n - 1, x] - (4*x^2 + (8*n - 20)*x^3)*D[P[n - 1, x], x] + 4*x^4*D[P[n - 1, x], {x, 2}]];
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] = (1 + (4*n - 6)*x + (4*n^2 - 16*n + 16)*x^2)*v[i - 1] - (4*x^2 + (8*n - 20)*x^3)*deriv(v[i - 1], x) + 4*x^4*deriv(deriv(v[i - 1], x), 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].?
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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[2,1].?
Related ?
Same fingerprint ?
- A079621 → θ+1
- A286724: Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].
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.