A269944 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-16 - 16\,x + 9\,x^{2} + 32\,n + 16\,n x - 24\,n^{2} - 4\,n^{2} x + 8\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(16\,x^{2} + 4\,x^{3} - 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(5 + 4\,x - 6\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 0 | 4 | 4 | ||||||
| 2 | 0 | 4 | 9 | 13 | |||||
| 3 | 0 | 16 | 45 | 16 | 77 | ||||
| 4 | 0 | 144 | 441 | 224 | 25 | 834 | |||
| 5 | 0 | 2304 | 7380 | 4368 | 750 | 36 | 14838 | ||
| 6 | 0 | 57600 | 189684 | 122320 | 25575 | 1980 | 49 | 397208 | |
| 7 | 0 | 2073600 | 6958224 | 4740736 | 1111825 | 108108 | 4459 | 64 | 14997016 |
Compute more
?
Technical details
- Node id
n-ab6e8549e7f75941· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-a25633ab4d4ed031
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: 0, 4; 2: 0, 4, 9; 3: 0, 16, 45, 16; 4: 0, 144, 441, 224, 25; 5: 0, 2304, 7380, 4368, 750, 36; 6: 0, 57600, 189684, 122320, 25575, 1980, 49; 7: 0, 2073600, 6958224, 4740736, 1111825, 108108, 4459, 64; 8: 0, 101606400, 345618576, 244666240, 61886825, 6898320, 365638, 8960, 81; 9: 0, 6502809600, 22348203264, 16273072384, 4343047800, 530609508, 32790212, 1051008, 16524, 100;
%F FormulaThe row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(4*n^2-16*n+16)*x-9*x^2)*P(n-2,x) + ((4*n^2-16*n+16)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+5+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^2 - 16*n + 16)*x - 9*x^2)*P[n - 2, x] + ((4*n^2 - 16*n + 16)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 5 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^2 - 16*n + 16)*x - 9*x^2)*v[i - 2] + ((4*n^2 - 16*n + 16)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 5 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], x)); 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
Proof chains ? JSON
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.