A185410 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-3 + 2\,x + n\right)\,P_{n}(x) = \left(6 - 34\,x + 96\,x^{2} - 84\,x^{3} + 16\,x^{4} - 3\,n + 22\,n x - 90\,n x^{2} + 64\,n x^{3} - 8\,n x^{4} - 4\,n^{2} x + 32\,n^{2} x^{2} - 12\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 32\,x^{2} + 60\,x^{3} - 40\,x^{4} + 8\,x^{5} - 2\,n x + 22\,n x^{2} - 32\,n x^{3} + 12\,n x^{4} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-11 + 38\,x - 28\,x^{2} + 4\,x^{3} + 4\,n - 22\,n x + 10\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 10\,x^{2} - 4\,x^{3} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||||||||
| 1 | 1 | 1 | |||||||||||||
| 2 | 3 | 2 | 5 | ||||||||||||
| 3 | 11 | 18 | 4 | 33 | |||||||||||
| 4 | 37 | 156 | 84 | 8 | 285 | ||||||||||
| 5 | 117 | 1148 | 1404 | 360 | 16 | 3045 | |||||||||
| 6 | 359 | 7374 | 19012 | 10480 | 1488 | 32 | 38745 | ||||||||
| 7 | 1087 | 43326 | 213840 | 236080 | 71280 | 6048 | 64 | 571725 | |||||||
| 8 | 3273 | 240536 | 2101272 | 4263120 | 2500880 | 460992 | 24384 | 128 | 9594585 | ||||||
| 9 | 9833 | 1287768 | 18788200 | 64905104 | 68339760 | 24079552 | 2896832 | 97920 | 256 | 180405225 | |||||
| 10 | 29515 | 6733818 | 157233432 | 871379712 | 1530598608 | 953531040 | 218283072 | 17895168 | 392448 | 512 | 3756077325 | ||||
| 11 | 88563 | 34672586 | 1255364268 | 10665914880 | 29445943632 | 30232855968 | 12121140864 | 1902557952 | 109397760 | 1571328 | 1024 | 85769508825 | |||
| 12 | 265709 | 176728308 | 9688505740 | 121891316760 | 504813242160 | 805270338432 | 528040498944 | 144436700160 | 16154845440 | 664407040 | 6288384 | 2048 | 2131143139125 | ||
| 13 | 797149 | 894801300 | 72942801156 | 1322792880440 | 7925026203360 | 18735582448512 | 18957644805120 | 8440766926848 | 1643917973760 | 134766832640 | 4018289664 | 25159680 | 4096 | 57238379923725 | |
| 14 | 2391471 | 4510675334 | 539226872508 | 13799860745712 | 116228401073920 | 392293338931008 | 584133991252992 | 399706697385984 | 126381219607296 | 18101130094080 | 1110682141696 | 24236519424 | 100651008 | 8192 | 1652323398350625 |
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Technical details
- Node id
n-81a73406b9e2b0b2· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-4516a549dfc2eb3d
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: 3, 2; 3: 11, 18, 4; 4: 37, 156, 84, 8; 5: 117, 1148, 1404, 360, 16; 6: 359, 7374, 19012, 10480, 1488, 32; 7: 1087, 43326, 213840, 236080, 71280, 6048, 64; 8: 3273, 240536, 2101272, 4263120, 2500880, 460992, 24384, 128; 9: 9833, 1287768, 18788200, 64905104, 68339760, 24079552, 2896832, 97920, 256;
%F FormulaThe row polynomials satisfy (n-3+2*x)*P(n,x) = -(3*n-6+(4*n^2-22*n+34)*x+(4*n^3-32*n^2+90*n-96)*x^2+(12*n^2-64*n+84)*x^3+(8*n-16)*x^4)*P(n-2,x) - ((2*n-4)*x+(4*n^2-22*n+32)*x^2-(4*n^2-32*n+60)*x^3-(12*n-40)*x^4-8*x^5)*d/dx P(n-2,x) + (4*n-11+(4*n^2-22*n+38)*x+(10*n-28)*x^2+4*x^3)*P(n-1,x) + ((2*n-6)*x-(2*n-10)*x^2-4*x^3)*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[(-(3*n - 6 + (4*n^2 - 22*n + 34)*x + (4*n^3 - 32*n^2 + 90*n - 96)*x^2 + (12*n^2 - 64*n + 84)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] - ((2*n - 4)*x + (4*n^2 - 22*n + 32)*x^2 - (4*n^2 - 32*n + 60)*x^3 - (12*n - 40)*x^4 - 8*x^5)*D[P[n - 2, x], x] + (4*n - 11 + (4*n^2 - 22*n + 38)*x + (10*n - 28)*x^2 + 4*x^3)*P[n - 1, x] + ((2*n - 6)*x - (2*n - 10)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n - 3 + 2*x)]];
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] = (-(3*n - 6 + (4*n^2 - 22*n + 34)*x + (4*n^3 - 32*n^2 + 90*n - 96)*x^2 + (12*n^2 - 64*n + 84)*x^3 + (8*n - 16)*x^4)*v[i - 2] - ((2*n - 4)*x + (4*n^2 - 22*n + 32)*x^2 - (4*n^2 - 32*n + 60)*x^3 - (12*n - 40)*x^4 - 8*x^5)*deriv(v[i - 2], x) + (4*n - 11 + (4*n^2 - 22*n + 38)*x + (10*n - 28)*x^2 + 4*x^3)*v[i - 1] + ((2*n - 6)*x - (2*n - 10)*x^2 - 4*x^3)*deriv(v[i - 1], x))/(n - 3 + 2*x)); 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 differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, 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 ≥ 1
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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.