A144442 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-21 + 20\,n\right)\,P_{n}(x) = \left(25\,x^{2} - 500\,n x^{2}\right)\,P_{n-4}(x) + \left(-40\,x + 35\,x^{2} + 800\,n x + 1850\,n x^{2} - 1000\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(25\,x^{2} - 25\,x^{3} - 500\,n x^{2} + 500\,n x^{3}\right)\,P_{n-3}'(x) + \left(-4 - 138\,x - 189\,x^{2} + 80\,n - 1280\,n x - 660\,n x^{2} + 800\,n^{2} x + 1325\,n^{2} x^{2} - 500\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-20\,x + 125\,x^{2} - 105\,x^{3} + 400\,n x + 25\,n x^{2} - 425\,n x^{3} - 500\,n^{2} x^{2} + 500\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-17 + 168\,x - 60\,n - 370\,n x + 200\,n^{2} x\right)\,P_{n-1}(x) + \left(-105\,x + 105\,x^{2} + 100\,n x - 100\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 34\,x + 3\,x^{2}\)
\(P_{3}(x) = 1 + 236\,x + 354\,x^{2} + 4\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 34 | 3 | 38 | |||||
| 3 | 1 | 236 | 354 | 4 | 595 | ||||
| 4 | 1 | 1458 | 8043 | 2916 | 5 | 12423 | |||
| 5 | 1 | 8800 | 125235 | 166980 | 22000 | 6 | 323022 | ||
| 6 | 1 | 52862 | 1665720 | 5396980 | 2776200 | 158586 | 7 | 10050356 | |
| 7 | 1 | 317244 | 20450538 | 133826740 | 167283425 | 40901076 | 1110354 | 8 | 363889386 |
Compute more
?
Technical details
- Node id
n-3cf14a762fddaaaa· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-0d6ccd89a44d9fdd
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: 1, 2; 2: 1, 34, 3; 3: 1, 236, 354, 4; 4: 1, 1458, 8043, 2916, 5; 5: 1, 8800, 125235, 166980, 22000, 6; 6: 1, 52862, 1665720, 5396980, 2776200, 158586, 7; 7: 1, 317244, 20450538, 133826740, 167283425, 40901076, 1110354, 8; 8: 1, 1903546, 240104229, 2861284624, 7059634975, 4291926936, 560243201, 7614184, 9; 9: 1, 11421368, 2746317333, 55841199036, 242080501880, 290496602256, 97722098313, 7323512888, 51396156, 10;
%F FormulaThe row polynomials satisfy (20*n-21)*P(n,x) = -(500*n-25)*x^2*P(n-4,x) + ((800*n-40)*x-(1000*n^2-1850*n-35)*x^2)*P(n-3,x) - ((500*n-25)*x^2-(500*n-25)*x^3)*d/dx P(n-3,x) + (80*n-4+(800*n^2-1280*n-138)*x-(500*n^3-1325*n^2+660*n+189)*x^2)*P(n-2,x) + ((400*n-20)*x-(500*n^2-25*n-125)*x^2+(500*n^2-425*n-105)*x^3)*d/dx P(n-2,x) - (60*n+17-(200*n^2-370*n+168)*x)*P(n-1,x) + ((100*n-105)*x-(100*n-105)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 34*x + 3*x^2, P(3,x) = 1 + 236*x + 354*x^2 + 4*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 34*x + 3*x^2;
P[3, x_] := 1 + 236*x + 354*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(500*n - 25)*x^2*P[n - 4, x] + ((800*n - 40)*x - (1000*n^2 - 1850*n - 35)*x^2)*P[n - 3, x] - ((500*n - 25)*x^2 - (500*n - 25)*x^3)*D[P[n - 3, x], x] + (80*n - 4 + (800*n^2 - 1280*n - 138)*x - (500*n^3 - 1325*n^2 + 660*n + 189)*x^2)*P[n - 2, x] + ((400*n - 20)*x - (500*n^2 - 25*n - 125)*x^2 + (500*n^2 - 425*n - 105)*x^3)*D[P[n - 2, x], x] - (60*n + 17 - (200*n^2 - 370*n + 168)*x)*P[n - 1, x] + ((100*n - 105)*x - (100*n - 105)*x^2)*D[P[n - 1, x], x])/(20*n - 21)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 34*x + 3*x^2; v[4] = 1 + 236*x + 354*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(500*n - 25)*x^2*v[i - 4] + ((800*n - 40)*x - (1000*n^2 - 1850*n - 35)*x^2)*v[i - 3] - ((500*n - 25)*x^2 - (500*n - 25)*x^3)*deriv(v[i - 3], x) + (80*n - 4 + (800*n^2 - 1280*n - 138)*x - (500*n^3 - 1325*n^2 + 660*n + 189)*x^2)*v[i - 2] + ((400*n - 20)*x - (500*n^2 - 25*n - 125)*x^2 + (500*n^2 - 425*n - 105)*x^3)*deriv(v[i - 2], x) - (60*n + 17 - (200*n^2 - 370*n + 168)*x)*v[i - 1] + ((100*n - 105)*x - (100*n - 105)*x^2)*deriv(v[i - 1], x))/(20*n - 21)); 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 multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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
- [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.