A134239 → basis A007318
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 8\,x + 2\,x^{2} - 4\,n - 4\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 4\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)
\(P_{2}(x) = 18 + 15\,x + 3\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 6 | 2 | 8 | ||||||
| 2 | 18 | 15 | 3 | 36 | |||||
| 3 | 48 | 64 | 28 | 4 | 144 | ||||
| 4 | 120 | 220 | 150 | 45 | 5 | 540 | |||
| 5 | 288 | 672 | 624 | 288 | 66 | 6 | 1944 | ||
| 6 | 672 | 1904 | 2240 | 1400 | 490 | 91 | 7 | 6804 | |
| 7 | 1536 | 5120 | 7296 | 5760 | 2720 | 768 | 120 | 8 | 23328 |
Compute more
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Technical details
- Node id
n-de4fe2266f386481· revision 0 · JSON- Definition source
derived:basis_map_A007318- Fingerprint
f6-1c35c8e273da035a
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 6, 2; 2: 18, 15, 3; 3: 48, 64, 28, 4; 4: 120, 220, 150, 45, 5; 5: 288, 672, 624, 288, 66, 6; 6: 672, 1904, 2240, 1400, 490, 91, 7; 7: 1536, 5120, 7296, 5760, 2720, 768, 120, 8; 8: 3456, 13248, 22176, 21168, 12600, 4788, 1134, 153, 9; 9: 7680, 33280, 64000, 71680, 51520, 24640, 7840, 1600, 190, 10;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 2*x, P(2,x) = 18 + 15*x + 3*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 6 + 2*x;
P[2, x_] := 18 + 15*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 2*x; v[3] = 18 + 15*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (4*n - 8 + (2*n - 4)*x)*v[i - 1])/(n - 2)); 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]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
A127899(unsigned) * A007318.?
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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
A127899(unsigned) * A007318.?
Related ?
Same fingerprint ?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.