A144442 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-4 + 5\,n\right)\,P_{n}(x) = \left(-25\,x^{2} - 125\,n x^{2}\right)\,P_{n-4}(x) + \left(-10\,x + 15\,x^{2} - 50\,n x + 650\,n x^{2} - 250\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-25\,x^{2} + 25\,x^{3} - 125\,n x^{2} + 125\,n x^{3}\right)\,P_{n-3}'(x) + \left(-6 - 12\,x + 224\,x^{2} - 30\,n + 180\,n x - 640\,n x^{2} - 50\,n^{2} x + 550\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - 75\,x^{2} + 80\,x^{3} - 25\,n x + 225\,n x^{2} - 200\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-23 + 72\,x + 35\,n - 130\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 20\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 17 + 2\,x\)
\(P_{3}(x) = 118 + 236\,x + 3\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 1 | 1 | ||||||
| 2 | 17 | 2 | 19 | |||||
| 3 | 118 | 236 | 3 | 357 | ||||
| 4 | 729 | 5362 | 2187 | 4 | 8282 | |||
| 5 | 4400 | 83490 | 125235 | 17600 | 5 | 230730 | ||
| 6 | 26431 | 1110480 | 4047735 | 2220960 | 132155 | 6 | 7537767 | |
| 7 | 158622 | 13633692 | 100370055 | 133826740 | 34084230 | 951732 | 7 | 283025078 |
Compute more
?
Technical details
- Node id
n-f7d4ac866eaf7630· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-e45b34e29a2fbcd1
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 1; 2: 17, 2; 3: 118, 236, 3; 4: 729, 5362, 2187, 4; 5: 4400, 83490, 125235, 17600, 5; 6: 26431, 1110480, 4047735, 2220960, 132155, 6; 7: 158622, 13633692, 100370055, 133826740, 34084230, 951732, 7; 8: 951773, 160069486, 2145963468, 5647707980, 3576605780, 480208458, 6662411, 8; 9: 5710684, 1830878222, 41880899277, 193664401504, 242080501880, 83761798554, 6408073777, 45685472, 9;
%F FormulaThe row polynomials satisfy (5*n-4)*P(n,x) = -(125*n+25)*x^2*P(n-4,x) - ((50*n+10)*x+(250*n^2-650*n-15)*x^2)*P(n-3,x) - ((125*n+25)*x^2-(125*n+25)*x^3)*d/dx P(n-3,x) - (30*n+6+(50*n^2-180*n+12)*x+(125*n^3-550*n^2+640*n-224)*x^2)*P(n-2,x) - ((25*n+5)*x+(125*n^2-225*n+75)*x^2-(125*n^2-200*n+80)*x^3)*d/dx P(n-2,x) + (35*n-23+(50*n^2-130*n+72)*x)*P(n-1,x) + ((25*n-20)*x-(25*n-20)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 17 + 2*x, P(3,x) = 118 + 236*x + 3*x^2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 17 + 2*x;
P[3, x_] := 118 + 236*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(125*n + 25)*x^2*P[n - 4, x] - ((50*n + 10)*x + (250*n^2 - 650*n - 15)*x^2)*P[n - 3, x] - ((125*n + 25)*x^2 - (125*n + 25)*x^3)*D[P[n - 3, x], x] - (30*n + 6 + (50*n^2 - 180*n + 12)*x + (125*n^3 - 550*n^2 + 640*n - 224)*x^2)*P[n - 2, x] - ((25*n + 5)*x + (125*n^2 - 225*n + 75)*x^2 - (125*n^2 - 200*n + 80)*x^3)*D[P[n - 2, x], x] + (35*n - 23 + (50*n^2 - 130*n + 72)*x)*P[n - 1, x] + ((25*n - 20)*x - (25*n - 20)*x^2)*D[P[n - 1, x], x])/(5*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 17 + 2*x; v[4] = 118 + 236*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(125*n + 25)*x^2*v[i - 4] - ((50*n + 10)*x + (250*n^2 - 650*n - 15)*x^2)*v[i - 3] - ((125*n + 25)*x^2 - (125*n + 25)*x^3)*deriv(v[i - 3], x) - (30*n + 6 + (50*n^2 - 180*n + 12)*x + (125*n^3 - 550*n^2 + 640*n - 224)*x^2)*v[i - 2] - ((25*n + 5)*x + (125*n^2 - 225*n + 75)*x^2 - (125*n^2 - 200*n + 80)*x^3)*deriv(v[i - 2], x) + (35*n - 23 + (50*n^2 - 130*n + 72)*x)*v[i - 1] + ((25*n - 20)*x - (25*n - 20)*x^2)*deriv(v[i - 1], x))/(5*n - 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
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
-
Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
-
-
from
Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
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.