A204420 → reverse
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-1 + 2\,x + 2\,n - 6\,n x + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 3 | 6 | 9 | |||||
| 3 | 15 | 90 | 120 | 225 | ||||
| 4 | 105 | 1260 | 4620 | 5040 | 11025 | |||
| 5 | 945 | 18900 | 132300 | 378000 | 362880 | 893025 | ||
| 6 | 10395 | 311850 | 3534300 | 18711000 | 45571680 | 39916800 | 108056025 | |
| 7 | 135135 | 5675670 | 94594500 | 794593800 | 3511347840 | 7628100480 | 6227020800 | 18261468225 |
Compute more
?
Technical details
- Node id
n-ed1a169c7e7a6346· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-181e40cb14f29d52
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 1; 2: 3, 6; 3: 15, 90, 120; 4: 105, 1260, 4620, 5040; 5: 945, 18900, 132300, 378000, 362880; 6: 10395, 311850, 3534300, 18711000, 45571680, 39916800; 7: 135135, 5675670, 94594500, 794593800, 3511347840, 7628100480, 6227020800; 8: 2027025, 113513400, 2610808200, 31783752000, 219534915600, 851804553600, 1695306412800, 1307674368000; 9: 34459425, 2481078600, 75259384200, 1250463614400, 12377274109200, 74194174454400, 260511047596800, 483353808537600, 355687428096000;
%F FormulaThe row polynomials satisfy P(n,x) = (2*n-1+(4*n^2-6*n+2)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + (4*n^2 - 6*n + 2)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + (4*n^2 - 6*n + 2)*x)*v[i - 1]); 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]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
Related ?
Same fingerprint ?
- A028338 → x ↦ x+1 → reverse
- A225471 → basis A013609 → reverse
- A028338 → x ↦ x+1
- A109692 → x ↦ x/(x+1)
- A204420: Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, …
- A204420 → basis A007318 → x ↦ x+1
- A225471 → basis A013609
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.