A257606 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(9 + 6\,n + n^{2}\right)\,P_{n}(x) = \left(-144 - 408\,x + 612\,x^{2} - 84\,n - 428\,n x + 876\,n x^{2} - 12\,n^{2} - 144\,n^{2} x + 461\,n^{2} x^{2} - 16\,n^{3} x + 106\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-111\,x + 642\,x^{2} - 171\,x^{3} - 56\,n x + 557\,n x^{2} - 24\,n x^{3} - 8\,n^{2} x + 164\,n^{2} x^{2} + 77\,n^{2} x^{3} + 16\,n^{3} x^{2} + 34\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-15\,x^{2} + 150\,x^{3} - 135\,x^{4} - 7\,n x^{2} + 133\,n x^{3} - 126\,n x^{4} - n^{2} x^{2} + 40\,n^{2} x^{3} - 39\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(72 + 108\,x + 45\,n + 108\,n x + 7\,n^{2} + 36\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(24\,x - 144\,x^{2} + 13\,n x - 132\,n x^{2} + 2\,n^{2} x - 40\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 16\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 4 | 16 | 20 | ||||||
| 2 | 16 | 160 | 144 | 320 | |||||
| 3 | 64 | 1184 | 2664 | 1024 | 4936 | ||||
| 4 | 256 | 7712 | 31968 | 30848 | 6400 | 77184 | |||
| 5 | 1024 | 46752 | 313272 | 556928 | 292200 | 36864 | 1247040 | ||
| 6 | 4096 | 270624 | 2721168 | 7796992 | 7558800 | 2435616 | 200704 | 20988000 | |
| 7 | 16384 | 1516960 | 21807144 | 93280000 | 145750000 | 87228576 | 18582760 | 1048576 | 369230400 |
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Technical details
- Node id
n-26fd66898404a56a· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-de45cc49490c28bd
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, 160, 144; 3: 64, 1184, 2664, 1024; 4: 256, 7712, 31968, 30848, 6400; 5: 1024, 46752, 313272, 556928, 292200, 36864; 6: 4096, 270624, 2721168, 7796992, 7558800, 2435616, 200704; 7: 16384, 1516960, 21807144, 93280000, 145750000, 87228576, 18582760, 1048576; 8: 65536, 8305696, 164974464, 1001874304, 2332000000, 2254217184, 898194304, 132891136, 5308416; 9: 262144, 44674208, 1195412760, 9945999488, 32744857400, 47152594656, 30459623432, 8500712960, 904652712, 26214400;
%F FormulaThe row polynomials satisfy (n^2+6*n+9)*P(n,x) = -(12*n^2+84*n+144+(16*n^3+144*n^2+428*n+408)*x-(9*n^4+106*n^3+461*n^2+876*n+612)*x^2)*P(n-2,x) - ((8*n^2+56*n+111)*x-(16*n^3+164*n^2+557*n+642)*x^2-(4*n^4+34*n^3+77*n^2-24*n-171)*x^3)*d/dx P(n-2,x) - ((n^2+7*n+15)*x^2-(4*n^3+40*n^2+133*n+150)*x^3+(4*n^3+39*n^2+126*n+135)*x^4)*d^2/dx^2 P(n-2,x) + (7*n^2+45*n+72+(4*n^3+36*n^2+108*n+108)*x)*P(n-1,x) + ((2*n^2+13*n+24)*x-(4*n^3+40*n^2+132*n+144)*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[(-(12*n^2 + 84*n + 144 + (16*n^3 + 144*n^2 + 428*n + 408)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*P[n - 2, x] - ((8*n^2 + 56*n + 111)*x - (16*n^3 + 164*n^2 + 557*n + 642)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*D[P[n - 2, x], x] - ((n^2 + 7*n + 15)*x^2 - (4*n^3 + 40*n^2 + 133*n + 150)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] + (7*n^2 + 45*n + 72 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*P[n - 1, x] + ((2*n^2 + 13*n + 24)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*D[P[n - 1, x], x])/(n^2 + 6*n + 9)]];
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] = (-(12*n^2 + 84*n + 144 + (16*n^3 + 144*n^2 + 428*n + 408)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*v[i - 2] - ((8*n^2 + 56*n + 111)*x - (16*n^3 + 164*n^2 + 557*n + 642)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*deriv(v[i - 2], x) - ((n^2 + 7*n + 15)*x^2 - (4*n^3 + 40*n^2 + 133*n + 150)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) + (7*n^2 + 45*n + 72 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*v[i - 1] + ((2*n^2 + 13*n + 24)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*deriv(v[i - 1], x))/(n^2 + 6*n + 9)); 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.