A060187 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-7 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(15 - 92\,x + 299\,x^{2} - 238\,x^{3} + 40\,x^{4} - 6\,n + 52\,n x - 250\,n x^{2} + 152\,n x^{3} - 16\,n x^{4} - 8\,n^{2} x + 76\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(10\,x - 88\,x^{2} + 154\,x^{3} - 92\,x^{4} + 16\,x^{5} - 4\,n x + 52\,n x^{2} - 72\,n x^{3} + 24\,n x^{4} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-26 + 100\,x - 66\,x^{2} + 8\,x^{3} + 8\,n - 52\,n x + 20\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-14\,x + 22\,x^{2} - 8\,x^{3} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 2 | 1 | 3 | |||||||||||||
| 3 | 7 | 8 | 1 | 16 | ||||||||||||
| 4 | 24 | 69 | 26 | 1 | 120 | |||||||||||
| 5 | 77 | 536 | 458 | 80 | 1 | 1152 | ||||||||||
| 6 | 238 | 3601 | 6728 | 2630 | 242 | 1 | 13440 | |||||||||
| 7 | 723 | 21808 | 81187 | 65720 | 14153 | 728 | 1 | 184320 | ||||||||
| 8 | 2180 | 123493 | 839826 | 1298615 | 563008 | 73731 | 2186 | 1 | 2903040 | |||||||
| 9 | 6553 | 669776 | 7787476 | 21185344 | 17101454 | 4474960 | 377476 | 6560 | 1 | 51609600 | ||||||
| 10 | 19674 | 3532353 | 66880256 | 299361148 | 416479764 | 199661126 | 34002352 | 1913724 | 19682 | 1 | 1021870080 | |||||
| 11 | 59039 | 18291320 | 544018853 | 3805994032 | 8542385622 | 6968632160 | 2155029610 | 251230032 | 9647483 | 59048 | 1 | 22295347200 | ||||
| 12 | 177136 | 93581989 | 4255944602 | 44750373117 | 153776922168 | 200116138698 | 104445781500 | 22061468898 | 1823402568 | 48473777 | 177146 | 1 | 531372441600 | |||
| 13 | 531429 | 474995368 | 32363965534 | 496244032016 | 2506486602327 | 4936751655408 | 4082432466132 | 1448090806080 | 217763980899 | 13081230888 | 243077678 | 531440 | 1 | 13733933875200 | ||
| 14 | 1594310 | 2398359697 | 241031131752 | 5263637587426 | 37849675695350 | 108219105453135 | 134544473307936 | 75304062940956 | 18974060482098 | 2095365831295 | 93126459320 | 1217514402 | 1594322 | 1 | 382588157952000 | |
| 15 | 4782955 | 12068325344 | 1767549420487 | 53955729563048 | 538546131994711 | 2162545961463320 | 3874709951485827 | 3247881992124816 | 1288284714087009 | 238462191081488 | 19801037300725 | 659553804104 | 6093949597 | 4782968 | 1 | 11426632984166400 |
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Technical details
- Node id
n-0e93089cb9384e11· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-04cae45d42f04794
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, 1; 3: 7, 8, 1; 4: 24, 69, 26, 1; 5: 77, 536, 458, 80, 1; 6: 238, 3601, 6728, 2630, 242, 1; 7: 723, 21808, 81187, 65720, 14153, 728, 1; 8: 2180, 123493, 839826, 1298615, 563008, 73731, 2186, 1; 9: 6553, 669776, 7787476, 21185344, 17101454, 4474960, 377476, 6560, 1; 10: 19674, 3532353, 66880256, 299361148, 416479764, 199661126, 34002352, 1913724, 19682, 1;
%F FormulaThe row polynomials satisfy (2*n-7+4*x)*P(n,x) = -(6*n-15+(8*n^2-52*n+92)*x+(8*n^3-76*n^2+250*n-299)*x^2+(24*n^2-152*n+238)*x^3+(16*n-40)*x^4)*P(n-2,x) - ((4*n-10)*x+(8*n^2-52*n+88)*x^2-(8*n^2-72*n+154)*x^3-(24*n-92)*x^4-16*x^5)*d/dx P(n-2,x) + (8*n-26+(8*n^2-52*n+100)*x+(20*n-66)*x^2+8*x^3)*P(n-1,x) + ((4*n-14)*x-(4*n-22)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 15 + (8*n^2 - 52*n + 92)*x + (8*n^3 - 76*n^2 + 250*n - 299)*x^2 + (24*n^2 - 152*n + 238)*x^3 + (16*n - 40)*x^4)*P[n - 2, x] - ((4*n - 10)*x + (8*n^2 - 52*n + 88)*x^2 - (8*n^2 - 72*n + 154)*x^3 - (24*n - 92)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (8*n - 26 + (8*n^2 - 52*n + 100)*x + (20*n - 66)*x^2 + 8*x^3)*P[n - 1, x] + ((4*n - 14)*x - (4*n - 22)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(2*n - 7 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(6*n - 15 + (8*n^2 - 52*n + 92)*x + (8*n^3 - 76*n^2 + 250*n - 299)*x^2 + (24*n^2 - 152*n + 238)*x^3 + (16*n - 40)*x^4)*v[i - 2] - ((4*n - 10)*x + (8*n^2 - 52*n + 88)*x^2 - (8*n^2 - 72*n + 154)*x^3 - (24*n - 92)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (8*n - 26 + (8*n^2 - 52*n + 100)*x + (20*n - 66)*x^2 + 8*x^3)*v[i - 1] + ((4*n - 14)*x - (4*n - 22)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(2*n - 7 + 4*x)); 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 differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
Related ?
Same fingerprint ?
References
- [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.