A209414 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-63\,x^{2} + 105\,x^{3} - 24\,x^{4} + 4\,n - 42\,n x + 205\,n x^{2} - 196\,n x^{3} + 32\,n x^{4} - 7\,n^{2} + 51\,n^{2} x - 144\,n^{2} x^{2} + 64\,n^{2} x^{3} + 3\,n^{3} - 20\,n^{3} x + 32\,n^{3} x^{2}\right)\,P_{n}(x) = \left(27\,x^{2} - 108\,x^{3} + 313\,x^{4} - 88\,x^{5} + n - 4\,n x - 59\,n x^{2} + 356\,n x^{3} - 516\,n x^{4} + 96\,n x^{5} - 4\,n^{2} - 16\,n^{2} x + 133\,n^{2} x^{2} - 368\,n^{2} x^{3} + 192\,n^{2} x^{4} + 3\,n^{3} - 2\,n^{3} x - 52\,n^{3} x^{2} + 96\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 2\,x^{3} - 149\,x^{4} + 208\,x^{5} - 64\,x^{6} + n x - 7\,n x^{2} - 90\,n x^{3} + 224\,n x^{4} - 128\,n x^{5} - 3\,n^{2} x - 13\,n^{2} x^{2} + 80\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 5 | 6 | 1 | 12 | |||||
| 3 | 8 | 25 | 12 | 1 | 46 | ||||
| 4 | 11 | 62 | 74 | 20 | 1 | 168 | |||
| 5 | 14 | 117 | 262 | 170 | 30 | 1 | 594 | ||
| 6 | 17 | 190 | 657 | 810 | 335 | 42 | 1 | 2052 | |
| 7 | 20 | 281 | 1340 | 2615 | 2058 | 595 | 56 | 1 | 6966 |
Compute more
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Technical details
- Node id
n-2ca3d7764b943c46· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-5646fc8ced9b0b71
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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 5, 6, 1; 3: 8, 25, 12, 1; 4: 11, 62, 74, 20, 1; 5: 14, 117, 262, 170, 30, 1; 6: 17, 190, 657, 810, 335, 42, 1; 7: 20, 281, 1340, 2615, 2058, 595, 56, 1; 8: 23, 390, 2392, 6584, 8322, 4564, 980, 72, 1; 9: 26, 517, 3894, 14040, 25436, 22590, 9156, 1524, 90, 1;
%F FormulaThe row polynomials satisfy (3*n^3-7*n^2+4*n-(20*n^3-51*n^2+42*n)*x+(32*n^3-144*n^2+205*n-63)*x^2+(64*n^2-196*n+105)*x^3+(32*n-24)*x^4)*P(n,x) = (3*n^3-4*n^2+n-(2*n^3+16*n^2+4*n)*x-(52*n^3-133*n^2+59*n-27)*x^2+(96*n^3-368*n^2+356*n-108)*x^3+(192*n^2-516*n+313)*x^4+(96*n-88)*x^5)*P(n-1,x) - ((3*n^2-n)*x+(13*n^2+7*n-3)*x^2-(80*n^2-90*n+2)*x^3+(64*n^2-224*n+149)*x^4+(128*n-208)*x^5+64*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 4*n^2 + n - (2*n^3 + 16*n^2 + 4*n)*x - (52*n^3 - 133*n^2 + 59*n - 27)*x^2 + (96*n^3 - 368*n^2 + 356*n - 108)*x^3 + (192*n^2 - 516*n + 313)*x^4 + (96*n - 88)*x^5)*P[n - 1, x] - ((3*n^2 - n)*x + (13*n^2 + 7*n - 3)*x^2 - (80*n^2 - 90*n + 2)*x^3 + (64*n^2 - 224*n + 149)*x^4 + (128*n - 208)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(3*n^3 - 7*n^2 + 4*n - (20*n^3 - 51*n^2 + 42*n)*x + (32*n^3 - 144*n^2 + 205*n - 63)*x^2 + (64*n^2 - 196*n + 105)*x^3 + (32*n - 24)*x^4)]];
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] = ((3*n^3 - 4*n^2 + n - (2*n^3 + 16*n^2 + 4*n)*x - (52*n^3 - 133*n^2 + 59*n - 27)*x^2 + (96*n^3 - 368*n^2 + 356*n - 108)*x^3 + (192*n^2 - 516*n + 313)*x^4 + (96*n - 88)*x^5)*v[i - 1] - ((3*n^2 - n)*x + (13*n^2 + 7*n - 3)*x^2 - (80*n^2 - 90*n + 2)*x^3 + (64*n^2 - 224*n + 149)*x^4 + (128*n - 208)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(3*n^3 - 7*n^2 + 4*n - (20*n^3 - 51*n^2 + 42*n)*x + (32*n^3 - 144*n^2 + 205*n - 63)*x^2 + (64*n^2 - 196*n + 105)*x^3 + (32*n - 24)*x^4)); 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
derived from Laguerre’s theorem on differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
References
- [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.