A257613 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(-16 - 24\,n - 80\,n x + 64\,n x^{2} - 8\,n^{2} - 96\,n^{2} x + 164\,n^{2} x^{2} - 32\,n^{3} x + 136\,n^{3} x^{2} + 36\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-44\,x + 120\,x^{2} - 92\,x^{3} - 48\,n x + 212\,n x^{2} - 208\,n x^{3} - 16\,n^{2} x + 144\,n^{2} x^{2} - 124\,n^{2} x^{3} + 32\,n^{3} x^{2} + 8\,n^{3} x^{3} + 16\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-12\,x^{2} + 40\,x^{3} - 28\,x^{4} - 12\,n x^{2} + 84\,n x^{3} - 72\,n x^{4} - 4\,n^{2} x^{2} + 64\,n^{2} x^{3} - 60\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(8 + 8\,x + 14\,n + 24\,n x + 6\,n^{2} + 24\,n^{2} x + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 16\,x^{2} + 10\,n x - 40\,n x^{2} + 4\,n^{2} x - 32\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 16\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 4 | 16 | 20 | ||||||
| 2 | 16 | 192 | 144 | 352 | |||||
| 3 | 64 | 1664 | 3744 | 1024 | 6496 | ||||
| 4 | 256 | 12544 | 59904 | 50176 | 6400 | 129280 | |||
| 5 | 1024 | 87552 | 761472 | 1353728 | 547200 | 36864 | 2787840 | ||
| 6 | 4096 | 582656 | 8455680 | 27074560 | 23488000 | 5243904 | 200704 | 65049600 | |
| 7 | 16384 | 3758080 | 85999104 | 451133440 | 704896000 | 343996416 | 46036480 | 1048576 | 1636884480 |
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Technical details
- Node id
n-c9e46b637e60b92e· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-5cf267279c5da1ec
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: 4, 16; 2: 16, 192, 144; 3: 64, 1664, 3744, 1024; 4: 256, 12544, 59904, 50176, 6400; 5: 1024, 87552, 761472, 1353728, 547200, 36864; 6: 4096, 582656, 8455680, 27074560, 23488000, 5243904, 200704; 7: 16384, 3758080, 85999104, 451133440, 704896000, 343996416, 46036480, 1048576; 8: 65536, 23728128, 823283712, 6651756544, 16917504000, 14966452224, 4482322432, 379650048, 5308416; 9: 262144, 147611648, 7547258880, 89935413248, 348517222400, 501864800256, 275427203072, 53669396480, 2989135872, 26214400;
%F FormulaThe row polynomials satisfy (n^2+2*n+1)*P(n,x) = -(8*n^2+24*n+16+(32*n^3+96*n^2+80*n)*x-(36*n^4+136*n^3+164*n^2+64*n)*x^2)*P(n-2,x) - ((16*n^2+48*n+44)*x-(32*n^3+144*n^2+212*n+120)*x^2-(16*n^4+8*n^3-124*n^2-208*n-92)*x^3)*d/dx P(n-2,x) - ((4*n^2+12*n+12)*x^2-(16*n^3+64*n^2+84*n+40)*x^3+(16*n^3+60*n^2+72*n+28)*x^4)*d^2/dx^2 P(n-2,x) + (6*n^2+14*n+8+(8*n^3+24*n^2+24*n+8)*x)*P(n-1,x) + ((4*n^2+10*n+8)*x-(8*n^3+32*n^2+40*n+16)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 16*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 4 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 + 24*n + 16 + (32*n^3 + 96*n^2 + 80*n)*x - (36*n^4 + 136*n^3 + 164*n^2 + 64*n)*x^2)*P[n - 2, x] - ((16*n^2 + 48*n + 44)*x - (32*n^3 + 144*n^2 + 212*n + 120)*x^2 - (16*n^4 + 8*n^3 - 124*n^2 - 208*n - 92)*x^3)*D[P[n - 2, x], x] - ((4*n^2 + 12*n + 12)*x^2 - (16*n^3 + 64*n^2 + 84*n + 40)*x^3 + (16*n^3 + 60*n^2 + 72*n + 28)*x^4)*D[P[n - 2, x], {x, 2}] + (6*n^2 + 14*n + 8 + (8*n^3 + 24*n^2 + 24*n + 8)*x)*P[n - 1, x] + ((4*n^2 + 10*n + 8)*x - (8*n^3 + 32*n^2 + 40*n + 16)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*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] = 4 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^2 + 24*n + 16 + (32*n^3 + 96*n^2 + 80*n)*x - (36*n^4 + 136*n^3 + 164*n^2 + 64*n)*x^2)*v[i - 2] - ((16*n^2 + 48*n + 44)*x - (32*n^3 + 144*n^2 + 212*n + 120)*x^2 - (16*n^4 + 8*n^3 - 124*n^2 - 208*n - 92)*x^3)*deriv(v[i - 2], x) - ((4*n^2 + 12*n + 12)*x^2 - (16*n^3 + 64*n^2 + 84*n + 40)*x^3 + (16*n^3 + 60*n^2 + 72*n + 28)*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n^2 + 14*n + 8 + (8*n^3 + 24*n^2 + 24*n + 8)*x)*v[i - 1] + ((4*n^2 + 10*n + 8)*x - (8*n^3 + 32*n^2 + 40*n + 16)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*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
Related ?
Same fingerprint ?
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.