A048854 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 - 14\,x + 80\,x^{2} + 8\,n x - 72\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 88\,x^{3} - 32\,n x^{3}\right)\,P_{n-1}'(x) + \left(16\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 2 | 3 | ||||||
| 3 | 1 | 12 | 12 | 25 | |||||
| 4 | 1 | 30 | 180 | 120 | 331 | ||||
| 5 | 1 | 56 | 840 | 3360 | 1680 | 5937 | |||
| 6 | 1 | 90 | 2520 | 25200 | 75600 | 30240 | 133651 | ||
| 7 | 1 | 132 | 5940 | 110880 | 831600 | 1995840 | 665280 | 3609673 | |
| 8 | 1 | 182 | 12012 | 360360 | 5045040 | 30270240 | 60540480 | 17297280 | 113525595 |
Compute more
?
Technical details
- Node id
n-a50309a7fde94c6d· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-14fed6d19df1fb72
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, 12, 12; 4: 1, 30, 180, 120; 5: 1, 56, 840, 3360, 1680; 6: 1, 90, 2520, 25200, 75600, 30240; 7: 1, 132, 5940, 110880, 831600, 1995840, 665280; 8: 1, 182, 12012, 360360, 5045040, 30270240, 60540480, 17297280; 9: 1, 240, 21840, 960960, 21621600, 242161920, 1210809600, 2075673600, 518918400; 10: 1, 306, 36720, 2227680, 73513440, 1323241920, 12350257920, 52929676800, 79394515200, 17643225600;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(8*n-14)*x+(16*n^2-72*n+80)*x^2)*P(n-1,x) - (8*x^2+(32*n-88)*x^3)*d/dx P(n-1,x) + 16*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 + (8*n - 14)*x + (16*n^2 - 72*n + 80)*x^2)*P[n - 1, x] - (8*x^2 + (32*n - 88)*x^3)*D[P[n - 1, x], x] + 16*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 + (8*n - 14)*x + (16*n^2 - 72*n + 80)*x^2)*v[i - 1] - (8*x^2 + (32*n - 88)*x^3)*deriv(v[i - 1], x) + 16*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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].?
-
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
-
from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].?
Related ?
Same fingerprint ?
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.