A225117 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(25 - 30\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(20 + 352\,x + 4800\,x^{2} - 42\,n - 1476\,n x - 12960\,n x^{2} + 18\,n^{2} + 1080\,n^{2} x + 12393\,n^{2} x^{2} - 216\,n^{3} x - 5022\,n^{3} x^{2} + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-75\,x - 310\,x^{2} + 4825\,x^{3} + 252\,n x + 405\,n x^{2} - 11940\,n x^{3} - 108\,n^{2} x - 540\,n^{2} x^{2} + 10017\,n^{2} x^{3} + 216\,n^{3} x^{2} - 3294\,n^{3} x^{3} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-63\,x^{2} - 762\,x^{3} + 825\,x^{4} + 189\,n x^{2} + 1701\,n x^{3} - 1890\,n x^{4} - 81\,n^{2} x^{2} - 1296\,n^{2} x^{3} + 1377\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(40 - 500\,x - 39\,n + 900\,n x + 9\,n^{2} - 540\,n^{2} x + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(96\,x + 200\,x^{2} - 153\,n x - 540\,n x^{2} + 54\,n^{2} x + 432\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 2 | 4 | 6 | ||||||
| 3 | 4 | 52 | 9 | 65 | |||||
| 4 | 8 | 372 | 540 | 16 | 936 | ||||
| 5 | 16 | 2180 | 10179 | 4016 | 25 | 16416 | |||
| 6 | 32 | 11732 | 130482 | 170848 | 25450 | 36 | 338580 | ||
| 7 | 64 | 60708 | 1387017 | 4199008 | 2224950 | 147204 | 49 | 8019000 | |
| 8 | 128 | 308404 | 13281624 | 78244592 | 96758800 | 24929964 | 802424 | 64 | 214326000 |
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Technical details
- Node id
n-7ee7d8e0be55305b· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-9cdd5dbabb33bf1f
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 -1)
- 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 2, 4; 3: 4, 52, 9; 4: 8, 372, 540, 16; 5: 16, 2180, 10179, 4016, 25; 6: 32, 11732, 130482, 170848, 25450, 36; 7: 64, 60708, 1387017, 4199008, 2224950, 147204, 49; 8: 128, 308404, 13281624, 78244592, 96758800, 24929964, 802424, 64; 9: 256, 1553284, 119437263, 1238478928, 2943966475, 1817136108, 253575637, 4193728, 81; 10: 512, 7792020, 1032385662, 17657593536, 72177503850, 86002366248, 29804754210, 2414861568, 21232854, 100;
%F FormulaThe row polynomials satisfy (9*n^2-30*n+25)*P(n,x) = (18*n^2-42*n+20-(216*n^3-1080*n^2+1476*n-352)*x+(729*n^4-5022*n^3+12393*n^2-12960*n+4800)*x^2)*P(n-2,x) - ((108*n^2-252*n+75)*x-(216*n^3-540*n^2+405*n-310)*x^2-(324*n^4-3294*n^3+10017*n^2-11940*n+4825)*x^3)*d/dx P(n-2,x) - ((81*n^2-189*n+63)*x^2-(324*n^3-1296*n^2+1701*n-762)*x^3+(324*n^3-1377*n^2+1890*n-825)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2-39*n+40+(108*n^3-540*n^2+900*n-500)*x)*P(n-1,x) + ((54*n^2-153*n+96)*x-(108*n^3-432*n^2+540*n-200)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 42*n + 20 - (216*n^3 - 1080*n^2 + 1476*n - 352)*x + (729*n^4 - 5022*n^3 + 12393*n^2 - 12960*n + 4800)*x^2)*P[n - 2, x] - ((108*n^2 - 252*n + 75)*x - (216*n^3 - 540*n^2 + 405*n - 310)*x^2 - (324*n^4 - 3294*n^3 + 10017*n^2 - 11940*n + 4825)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 189*n + 63)*x^2 - (324*n^3 - 1296*n^2 + 1701*n - 762)*x^3 + (324*n^3 - 1377*n^2 + 1890*n - 825)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 - 39*n + 40 + (108*n^3 - 540*n^2 + 900*n - 500)*x)*P[n - 1, x] + ((54*n^2 - 153*n + 96)*x - (108*n^3 - 432*n^2 + 540*n - 200)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 30*n + 25)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((18*n^2 - 42*n + 20 - (216*n^3 - 1080*n^2 + 1476*n - 352)*x + (729*n^4 - 5022*n^3 + 12393*n^2 - 12960*n + 4800)*x^2)*v[i - 2] - ((108*n^2 - 252*n + 75)*x - (216*n^3 - 540*n^2 + 405*n - 310)*x^2 - (324*n^4 - 3294*n^3 + 10017*n^2 - 11940*n + 4825)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 189*n + 63)*x^2 - (324*n^3 - 1296*n^2 + 1701*n - 762)*x^3 + (324*n^3 - 1377*n^2 + 1890*n - 825)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 - 39*n + 40 + (108*n^3 - 540*n^2 + 900*n - 500)*x)*v[i - 1] + ((54*n^2 - 153*n + 96)*x - (108*n^3 - 432*n^2 + 540*n - 200)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 30*n + 25)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, 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 ≥ 2
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.