A185410 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-16\,x + 16\,x^{2} - n - 8\,n x - 112\,n x^{2} + n^{2} + 24\,n^{2} x + 212\,n^{2} x^{2} - 8\,n^{3} x - 152\,n^{3} x^{2} + 36\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(3\,x - 14\,x^{2} + 20\,x^{3} + 4\,n x - 4\,n x^{2} - 128\,n x^{3} - 4\,n^{2} x + 212\,n^{2} x^{3} + 8\,n^{3} x^{2} - 120\,n^{3} x^{3} + 16\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - 6\,x^{3} + 4\,x^{4} + 4\,n x^{2} + 20\,n x^{3} - 24\,n x^{4} - 4\,n^{2} x^{2} - 32\,n^{2} x^{3} + 36\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(1 - 8\,x - n + 24\,n x - 24\,n^{2} x + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 6\,n x - 8\,n x^{2} + 4\,n^{2} x + 16\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 1 | 8 | 9 | |||||
| 3 | 1 | 40 | 36 | 77 | ||||
| 4 | 1 | 144 | 540 | 128 | 813 | |||
| 5 | 1 | 464 | 4644 | 4736 | 400 | 10245 | ||
| 6 | 1 | 1432 | 31572 | 82688 | 33200 | 1152 | 150045 | |
| 7 | 1 | 4344 | 190080 | 1027840 | 1074000 | 203904 | 3136 | 2503305 |
Compute more
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Technical details
- Node id
n-6cf44b40f7f67fc8· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-1da90f686ef9e4c9
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 1, 8; 3: 1, 40, 36; 4: 1, 144, 540, 128; 5: 1, 464, 4644, 4736, 400; 6: 1, 1432, 31572, 82688, 33200, 1152; 7: 1, 4344, 190080, 1027840, 1074000, 203904, 3136; 8: 1, 13088, 1067688, 10574080, 22514000, 11522304, 1150912, 8192; 9: 1, 39328, 5750712, 96795904, 367846000, 386106624, 109060672, 6135808, 20736;
%F FormulaThe row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2-n-(8*n^3-24*n^2+8*n+16)*x+(36*n^4-152*n^3+212*n^2-112*n+16)*x^2)*P(n-2,x) - ((4*n^2-4*n-3)*x-(8*n^3-4*n-14)*x^2-(16*n^4-120*n^3+212*n^2-128*n+20)*x^3)*d/dx P(n-2,x) - ((4*n^2-4*n-2)*x^2-(16*n^3-32*n^2+20*n-6)*x^3+(16*n^3-36*n^2+24*n-4)*x^4)*d^2/dx^2 P(n-2,x) - (n-1-(8*n^3-24*n^2+24*n-8)*x)*P(n-1,x) + ((4*n^2-6*n+1)*x-(8*n^3-16*n^2+8*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n - (8*n^3 - 24*n^2 + 8*n + 16)*x + (36*n^4 - 152*n^3 + 212*n^2 - 112*n + 16)*x^2)*P[n - 2, x] - ((4*n^2 - 4*n - 3)*x - (8*n^3 - 4*n - 14)*x^2 - (16*n^4 - 120*n^3 + 212*n^2 - 128*n + 20)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4*n - 2)*x^2 - (16*n^3 - 32*n^2 + 20*n - 6)*x^3 + (16*n^3 - 36*n^2 + 24*n - 4)*x^4)*D[P[n - 2, x], {x, 2}] - (n - 1 - (8*n^3 - 24*n^2 + 24*n - 8)*x)*P[n - 1, x] + ((4*n^2 - 6*n + 1)*x - (8*n^3 - 16*n^2 + 8*n)*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - n - (8*n^3 - 24*n^2 + 8*n + 16)*x + (36*n^4 - 152*n^3 + 212*n^2 - 112*n + 16)*x^2)*v[i - 2] - ((4*n^2 - 4*n - 3)*x - (8*n^3 - 4*n - 14)*x^2 - (16*n^4 - 120*n^3 + 212*n^2 - 128*n + 20)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4*n - 2)*x^2 - (16*n^3 - 32*n^2 + 20*n - 6)*x^3 + (16*n^3 - 36*n^2 + 24*n - 4)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - 1 - (8*n^3 - 24*n^2 + 24*n - 8)*x)*v[i - 1] + ((4*n^2 - 6*n + 1)*x - (8*n^3 - 16*n^2 + 8*n)*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
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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
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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.