A257610 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 - 6\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-4 - 208\,x - 48\,x^{2} + 6\,n + 252\,n x + 216\,n x^{2} + 18\,n^{2} + 216\,n^{2} x + 81\,n^{2} x^{2} - 216\,n^{3} x - 1134\,n^{3} x^{2} + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(69\,x - 218\,x^{2} - 71\,x^{3} - 36\,n x + 117\,n x^{2} + 276\,n x^{3} - 108\,n^{2} x + 324\,n^{2} x^{2} + 297\,n^{2} x^{3} + 216\,n^{3} x^{2} - 1566\,n^{3} x^{3} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(45\,x^{2} - 30\,x^{3} - 15\,x^{4} - 27\,n x^{2} - 27\,n x^{3} + 54\,n x^{4} - 81\,n^{2} x^{2} + 81\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(4 - 4\,x - 15\,n + 36\,n x + 9\,n^{2} - 108\,n^{2} x + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(-12\,x - 8\,x^{2} - 9\,n x + 36\,n x^{2} + 54\,n^{2} x - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 8 | 10 | ||||||
| 2 | 4 | 80 | 36 | 120 | |||||
| 3 | 8 | 528 | 1188 | 128 | 1852 | ||||
| 4 | 16 | 2992 | 19008 | 11968 | 400 | 34384 | |||
| 5 | 32 | 15856 | 226116 | 401984 | 99100 | 1152 | 744240 | ||
| 6 | 64 | 81456 | 2308392 | 8843648 | 6412200 | 733104 | 3136 | 18382000 | |
| 7 | 128 | 412400 | 21582828 | 154733440 | 241771000 | 86331312 | 5051900 | 8192 | 509891200 |
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Technical details
- Node id
n-f882fbd3dc52436c· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-344812a88580a3fd
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, 8; 2: 4, 80, 36; 3: 8, 528, 1188, 128; 4: 16, 2992, 19008, 11968, 400; 5: 32, 15856, 226116, 401984, 99100, 1152; 6: 64, 81456, 2308392, 8843648, 6412200, 733104, 3136; 7: 128, 412400, 21582828, 154733440, 241771000, 86331312, 5051900, 8192; 8: 256, 2073776, 191220624, 2354348864, 6769588000, 5297284944, 1041090064, 33180416, 20736; 9: 512, 10395504, 1637082900, 32696793024, 157311623700, 226528738128, 100133928636, 11641478400, 210508956, 51200;
%F FormulaThe row polynomials satisfy (9*n^2-6*n+1)*P(n,x) = (18*n^2+6*n-4-(216*n^3-216*n^2-252*n+208)*x+(729*n^4-1134*n^3+81*n^2+216*n-48)*x^2)*P(n-2,x) - ((108*n^2+36*n-69)*x-(216*n^3+324*n^2+117*n-218)*x^2-(324*n^4-1566*n^3+297*n^2+276*n-71)*x^3)*d/dx P(n-2,x) - ((81*n^2+27*n-45)*x^2-(324*n^3-27*n-30)*x^3+(324*n^3-81*n^2-54*n+15)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2-15*n+4+(108*n^3-108*n^2+36*n-4)*x)*P(n-1,x) + ((54*n^2-9*n-12)*x-(108*n^3-36*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 + 6*n - 4 - (216*n^3 - 216*n^2 - 252*n + 208)*x + (729*n^4 - 1134*n^3 + 81*n^2 + 216*n - 48)*x^2)*P[n - 2, x] - ((108*n^2 + 36*n - 69)*x - (216*n^3 + 324*n^2 + 117*n - 218)*x^2 - (324*n^4 - 1566*n^3 + 297*n^2 + 276*n - 71)*x^3)*D[P[n - 2, x], x] - ((81*n^2 + 27*n - 45)*x^2 - (324*n^3 - 27*n - 30)*x^3 + (324*n^3 - 81*n^2 - 54*n + 15)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 - 15*n + 4 + (108*n^3 - 108*n^2 + 36*n - 4)*x)*P[n - 1, x] + ((54*n^2 - 9*n - 12)*x - (108*n^3 - 36*n + 8)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 6*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^2 + 6*n - 4 - (216*n^3 - 216*n^2 - 252*n + 208)*x + (729*n^4 - 1134*n^3 + 81*n^2 + 216*n - 48)*x^2)*v[i - 2] - ((108*n^2 + 36*n - 69)*x - (216*n^3 + 324*n^2 + 117*n - 218)*x^2 - (324*n^4 - 1566*n^3 + 297*n^2 + 276*n - 71)*x^3)*deriv(v[i - 2], x) - ((81*n^2 + 27*n - 45)*x^2 - (324*n^3 - 27*n - 30)*x^3 + (324*n^3 - 81*n^2 - 54*n + 15)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 - 15*n + 4 + (108*n^3 - 108*n^2 + 36*n - 4)*x)*v[i - 1] + ((54*n^2 - 9*n - 12)*x - (108*n^3 - 36*n + 8)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 6*n + 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
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
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.