A257607 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(16 + 8\,n + n^{2}\right)\,P_{n}(x) = \left(-400 - 1248\,x + 2064\,x^{2} - 180\,n - 956\,n x + 2152\,n x^{2} - 20\,n^{2} - 240\,n^{2} x + 833\,n^{2} x^{2} - 20\,n^{3} x + 142\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-224\,x + 1704\,x^{2} - 80\,x^{3} - 90\,n x + 1157\,n x^{2} + 248\,n x^{3} - 10\,n^{2} x + 264\,n^{2} x^{2} + 203\,n^{2} x^{3} + 20\,n^{3} x^{2} + 50\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-24\,x^{2} + 328\,x^{3} - 304\,x^{4} - 9\,n x^{2} + 225\,n x^{3} - 216\,n x^{4} - n^{2} x^{2} + 52\,n^{2} x^{3} - 51\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(160 + 256\,x + 76\,n + 192\,n x + 9\,n^{2} + 48\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(40\,x - 320\,x^{2} + 17\,n x - 224\,n x^{2} + 2\,n^{2} x - 52\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 20\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 5 | 20 | 25 | ||||||
| 2 | 25 | 240 | 225 | 490 | |||||
| 3 | 125 | 2140 | 4815 | 2000 | 9080 | ||||
| 4 | 625 | 16840 | 67410 | 67360 | 15625 | 167860 | |||
| 5 | 3125 | 123540 | 774990 | 1377760 | 772125 | 112500 | 3164040 | ||
| 6 | 15625 | 866240 | 7926615 | 22044160 | 22018375 | 7796160 | 765625 | 61432800 | |
| 7 | 78125 | 5884940 | 74976705 | 303179120 | 473717375 | 299906820 | 72090515 | 5000000 | 1234833600 |
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Technical details
- Node id
n-7ff8ccca84d0b5fc· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-92bc0d5de9f0211c
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: 5, 20; 2: 25, 240, 225; 3: 125, 2140, 4815, 2000; 4: 625, 16840, 67410, 67360, 15625; 5: 3125, 123540, 774990, 1377760, 772125, 112500; 6: 15625, 866240, 7926615, 22044160, 22018375, 7796160, 765625; 7: 78125, 5884940, 74976705, 303179120, 473717375, 299906820, 72090515, 5000000; 8: 390625, 39059640, 670489200, 3758352160, 8526912750, 8456292360, 3650441200, 624954240, 31640625; 9: 1953125, 254670340, 5748034680, 43178606080, 135466467250, 195071712840, 132234481120, 40874913280, 5157074385, 195312500;
%F FormulaThe row polynomials satisfy (n^2+8*n+16)*P(n,x) = -(20*n^2+180*n+400+(20*n^3+240*n^2+956*n+1248)*x-(9*n^4+142*n^3+833*n^2+2152*n+2064)*x^2)*P(n-2,x) - ((10*n^2+90*n+224)*x-(20*n^3+264*n^2+1157*n+1704)*x^2-(4*n^4+50*n^3+203*n^2+248*n-80)*x^3)*d/dx P(n-2,x) - ((n^2+9*n+24)*x^2-(4*n^3+52*n^2+225*n+328)*x^3+(4*n^3+51*n^2+216*n+304)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2+76*n+160+(4*n^3+48*n^2+192*n+256)*x)*P(n-1,x) + ((2*n^2+17*n+40)*x-(4*n^3+52*n^2+224*n+320)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 20*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 5 + 20*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n^2 + 180*n + 400 + (20*n^3 + 240*n^2 + 956*n + 1248)*x - (9*n^4 + 142*n^3 + 833*n^2 + 2152*n + 2064)*x^2)*P[n - 2, x] - ((10*n^2 + 90*n + 224)*x - (20*n^3 + 264*n^2 + 1157*n + 1704)*x^2 - (4*n^4 + 50*n^3 + 203*n^2 + 248*n - 80)*x^3)*D[P[n - 2, x], x] - ((n^2 + 9*n + 24)*x^2 - (4*n^3 + 52*n^2 + 225*n + 328)*x^3 + (4*n^3 + 51*n^2 + 216*n + 304)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 + 76*n + 160 + (4*n^3 + 48*n^2 + 192*n + 256)*x)*P[n - 1, x] + ((2*n^2 + 17*n + 40)*x - (4*n^3 + 52*n^2 + 224*n + 320)*x^2)*D[P[n - 1, x], x])/(n^2 + 8*n + 16)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n^2 + 180*n + 400 + (20*n^3 + 240*n^2 + 956*n + 1248)*x - (9*n^4 + 142*n^3 + 833*n^2 + 2152*n + 2064)*x^2)*v[i - 2] - ((10*n^2 + 90*n + 224)*x - (20*n^3 + 264*n^2 + 1157*n + 1704)*x^2 - (4*n^4 + 50*n^3 + 203*n^2 + 248*n - 80)*x^3)*deriv(v[i - 2], x) - ((n^2 + 9*n + 24)*x^2 - (4*n^3 + 52*n^2 + 225*n + 328)*x^3 + (4*n^3 + 51*n^2 + 216*n + 304)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 + 76*n + 160 + (4*n^3 + 48*n^2 + 192*n + 256)*x)*v[i - 1] + ((2*n^2 + 17*n + 40)*x - (4*n^3 + 52*n^2 + 224*n + 320)*x^2)*deriv(v[i - 1], x))/(n^2 + 8*n + 16)); 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.