A142459 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-7 + 4\,n\right)\,P_{n}(x) = \left(-9 + 198\,x + 231\,x^{2} + 12\,n - 336\,n x - 524\,n x^{2} + 96\,n^{2} x + 336\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-36\,x - 48\,x^{2} + 84\,x^{3} + 48\,n x + 112\,n x^{2} - 160\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 98\,x - 8\,n - 112\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-28\,x + 28\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 1 | 2 | 3 | |||||||||||||
| 3 | 1 | 20 | 3 | 24 | ||||||||||||
| 4 | 1 | 118 | 177 | 4 | 300 | |||||||||||
| 5 | 1 | 616 | 3186 | 1232 | 5 | 5040 | ||||||||||
| 6 | 1 | 3114 | 40686 | 54248 | 7785 | 6 | 105840 | |||||||||
| 7 | 1 | 15612 | 445581 | 1410448 | 742635 | 46836 | 7 | 2661120 | ||||||||
| 8 | 1 | 78110 | 4502007 | 28435660 | 35544575 | 9004014 | 273385 | 8 | 77837760 | |||||||
| 9 | 1 | 390608 | 43447188 | 495719776 | 1208515550 | 743579664 | 101376772 | 1562432 | 9 | 2594592000 | ||||||
| 10 | 1 | 1953106 | 408016140 | 7892596688 | 33557408470 | 40268890164 | 13812044204 | 1088043040 | 8788977 | 10 | 97037740800 | |||||
| 11 | 1 | 9765604 | 3768824007 | 118380381024 | 817119590490 | 1691293386888 | 1143967426686 | 236760762048 | 11306472021 | 48828020 | 11 | 4022655436800 | ||||
| 12 | 1 | 48828102 | 34461407085 | 1704773209620 | 18182321850450 | 60030748839348 | 70035873645906 | 29091714960720 | 3835739721645 | 114871356950 | 268554561 | 12 | 183030822374400 | |||
| 13 | 1 | 244140600 | 313155592038 | 23862147807920 | 379421366354475 | 1893390526021968 | 3501793682295300 | 2524520701362624 | 682958459438055 | 59655369519800 | 1148237170806 | 1464843600 | 13 | 9067065354547200 | ||
| 14 | 1 | 1220703098 | 2834879818842 | 327327093867704 | 7553787564142375 | 54786287154098538 | 151604554854459084 | 173262348405096096 | 82179430731147807 | 15107575128284750 | 900149508136186 | 11339519275368 | 7934570137 | 14 | 485735643993600000 | |
| 15 | 1 | 6103515596 | 25603640047281 | 4425345009410672 | 145189902151140205 | 1485900198083990748 | 5899385926794270813 | 10049216207495573568 | 7584924763021205331 | 2476500330139984580 | 319417784732508451 | 13276035028232016 | 110949106871551 | 42724609172 | 15 | 27978373094031360000 |
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Technical details
- Node id
n-c3dfc3a32f923c60· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-b99dd02e1a9529cf
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: 1, 2; 3: 1, 20, 3; 4: 1, 118, 177, 4; 5: 1, 616, 3186, 1232, 5; 6: 1, 3114, 40686, 54248, 7785, 6; 7: 1, 15612, 445581, 1410448, 742635, 46836, 7; 8: 1, 78110, 4502007, 28435660, 35544575, 9004014, 273385, 8; 9: 1, 390608, 43447188, 495719776, 1208515550, 743579664, 101376772, 1562432, 9; 10: 1, 1953106, 408016140, 7892596688, 33557408470, 40268890164, 13812044204, 1088043040, 8788977, 10;
%F FormulaThe row polynomials satisfy (4*n-7)*P(n,x) = (12*n-9+(96*n^2-336*n+198)*x-(64*n^3-336*n^2+524*n-231)*x^2)*P(n-2,x) + ((48*n-36)*x-(64*n^2-112*n+48)*x^2+(64*n^2-160*n+84)*x^3)*d/dx P(n-2,x) - (8*n-2-(32*n^2-112*n+98)*x)*P(n-1,x) + ((16*n-28)*x-(16*n-28)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n - 9 + (96*n^2 - 336*n + 198)*x - (64*n^3 - 336*n^2 + 524*n - 231)*x^2)*P[n - 2, x] + ((48*n - 36)*x - (64*n^2 - 112*n + 48)*x^2 + (64*n^2 - 160*n + 84)*x^3)*D[P[n - 2, x], x] - (8*n - 2 - (32*n^2 - 112*n + 98)*x)*P[n - 1, x] + ((16*n - 28)*x - (16*n - 28)*x^2)*D[P[n - 1, x], x])/(4*n - 7)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = ((12*n - 9 + (96*n^2 - 336*n + 198)*x - (64*n^3 - 336*n^2 + 524*n - 231)*x^2)*v[i - 2] + ((48*n - 36)*x - (64*n^2 - 112*n + 48)*x^2 + (64*n^2 - 160*n + 84)*x^3)*deriv(v[i - 2], x) - (8*n - 2 - (32*n^2 - 112*n + 98)*x)*v[i - 1] + ((16*n - 28)*x - (16*n - 28)*x^2)*deriv(v[i - 1], x))/(4*n - 7)); 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
Related ?
Same fingerprint ?
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.