A144440 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(8\right)\,P_{n}(x) = \left(96 - 60\,x - 126\,x^{2} - 252\,x^{3} - 162\,n - 81\,n x + 324\,n x^{3} + 54\,n^{2} + 27\,n^{2} x - 81\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(48\,x - 54\,x^{2} - 252\,x^{3} - 81\,n x + 324\,n x^{3} + 27\,n^{2} x - 81\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-336 + 586\,x + 480\,x^{2} + 528\,x^{3} + 663\,n - 114\,n x - 90\,n x^{2} - 1116\,n x^{3} - 351\,n^{2} - 99\,n^{2} x - 27\,n^{2} x^{2} + 540\,n^{2} x^{3} + 54\,n^{3} + 27\,n^{3} x - 81\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(112 - 398\,x + 174\,x^{2} + 978\,x^{3} + 252\,x^{4} - 189\,n + 507\,n x + 135\,n x^{2} - 1440\,n x^{3} - 324\,n x^{4} + 63\,n^{2} - 234\,n^{2} x - 54\,n^{2} x^{2} + 621\,n^{2} x^{3} + 81\,n^{2} x^{4} + 27\,n^{3} x - 81\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(48\,x - 102\,x^{2} - 198\,x^{3} + 252\,x^{4} - 81\,n x + 81\,n x^{2} + 324\,n x^{3} - 324\,n x^{4} + 27\,n^{2} x - 27\,n^{2} x^{2} - 81\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(48 - 454\,x - 344\,x^{2} + 240\,x^{3} - 45\,n + 582\,n x + 240\,n x^{2} - 144\,n x^{3} - 9\,n^{2} - 108\,n^{2} x - 45\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-16 + 234\,x - 240\,x^{2} - 36\,x^{3} + 144\,x^{4} + 27\,n - 54\,n x - 54\,n x^{2} - 9\,n^{2} + 27\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(72\,x^{2} - 144\,x^{3} + 72\,x^{4}\right)\,P_{n-2}''(x) + \left(16 - 164\,x + 48\,x^{2} + 18\,n + 48\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 12 + 13\,x + x^{2}\)
\(P_{3}(x) = 55 + 162\,x + 57\,x^{2} + x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 12 | 13 | 1 | 26 | |||||
| 3 | 55 | 162 | 57 | 1 | 275 | ||||
| 4 | 230 | 1807 | 1476 | 233 | 1 | 3747 | |||
| 5 | 933 | 16882 | 31900 | 11703 | 937 | 1 | 62356 | ||
| 6 | 3748 | 141003 | 554229 | 436379 | 87363 | 3753 | 1 | 1226476 | |
| 7 | 15011 | 1101298 | 8147421 | 12673795 | 5250479 | 633204 | 15017 | 1 | 27836226 |
Compute more
?
Technical details
- Node id
n-b493569528b7b7bc· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-16611f922ff0a6c9
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: 2, 1; 2: 12, 13, 1; 3: 55, 162, 57, 1; 4: 230, 1807, 1476, 233, 1; 5: 933, 16882, 31900, 11703, 937, 1; 6: 3748, 141003, 554229, 436379, 87363, 3753, 1; 7: 15011, 1101298, 8147421, 12673795, 5250479, 633204, 15017, 1; 8: 60066, 8256943, 106829773, 299801118, 237608225, 58834412, 4518894, 60073, 1; 9: 240289, 60351354, 1295868192, 6096201873, 8511396480, 3950191398, 632326284, 31977837, 240297, 1;
%F FormulaThe row polynomials satisfy 8*P(n,x) = (54*n^2-162*n+96+(27*n^2-81*n-60)*x-126*x^2-(81*n^2-324*n+252)*x^3)*P(n-4,x) + ((27*n^2-81*n+48)*x-54*x^2-(81*n^2-324*n+252)*x^3)*d/dx P(n-4,x) + (54*n^3-351*n^2+663*n-336+(27*n^3-99*n^2-114*n+586)*x-(27*n^2+90*n-480)*x^2-(81*n^3-540*n^2+1116*n-528)*x^3)*P(n-3,x) + (63*n^2-189*n+112+(27*n^3-234*n^2+507*n-398)*x-(54*n^2-135*n-174)*x^2-(81*n^3-621*n^2+1440*n-978)*x^3+(81*n^2-324*n+252)*x^4)*d/dx P(n-3,x) + ((27*n^2-81*n+48)*x-(27*n^2-81*n+102)*x^2-(81*n^2-324*n+198)*x^3+(81*n^2-324*n+252)*x^4)*d^2/dx^2 P(n-3,x) - (9*n^2+45*n-48+(108*n^2-582*n+454)*x+(45*n^2-240*n+344)*x^2+(144*n-240)*x^3)*P(n-2,x) - (9*n^2-27*n+16+(54*n-234)*x-(27*n^2-54*n-240)*x^2+36*x^3-144*x^4)*d/dx P(n-2,x) + (72*x^2-144*x^3+72*x^4)*d^2/dx^2 P(n-2,x) + (18*n+16+(48*n-164)*x+48*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 12 + 13*x + x^2, P(3,x) = 55 + 162*x + 57*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 12 + 13*x + x^2;
P[3, x_] := 55 + 162*x + 57*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((54*n^2 - 162*n + 96 + (27*n^2 - 81*n - 60)*x - 126*x^2 - (81*n^2 - 324*n + 252)*x^3)*P[n - 4, x] + ((27*n^2 - 81*n + 48)*x - 54*x^2 - (81*n^2 - 324*n + 252)*x^3)*D[P[n - 4, x], x] + (54*n^3 - 351*n^2 + 663*n - 336 + (27*n^3 - 99*n^2 - 114*n + 586)*x - (27*n^2 + 90*n - 480)*x^2 - (81*n^3 - 540*n^2 + 1116*n - 528)*x^3)*P[n - 3, x] + (63*n^2 - 189*n + 112 + (27*n^3 - 234*n^2 + 507*n - 398)*x - (54*n^2 - 135*n - 174)*x^2 - (81*n^3 - 621*n^2 + 1440*n - 978)*x^3 + (81*n^2 - 324*n + 252)*x^4)*D[P[n - 3, x], x] + ((27*n^2 - 81*n + 48)*x - (27*n^2 - 81*n + 102)*x^2 - (81*n^2 - 324*n + 198)*x^3 + (81*n^2 - 324*n + 252)*x^4)*D[P[n - 3, x], {x, 2}] - (9*n^2 + 45*n - 48 + (108*n^2 - 582*n + 454)*x + (45*n^2 - 240*n + 344)*x^2 + (144*n - 240)*x^3)*P[n - 2, x] - (9*n^2 - 27*n + 16 + (54*n - 234)*x - (27*n^2 - 54*n - 240)*x^2 + 36*x^3 - 144*x^4)*D[P[n - 2, x], x] + (72*x^2 - 144*x^3 + 72*x^4)*D[P[n - 2, x], {x, 2}] + (18*n + 16 + (48*n - 164)*x + 48*x^2)*P[n - 1, x])/(8)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 12 + 13*x + x^2; v[4] = 55 + 162*x + 57*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((54*n^2 - 162*n + 96 + (27*n^2 - 81*n - 60)*x - 126*x^2 - (81*n^2 - 324*n + 252)*x^3)*v[i - 4] + ((27*n^2 - 81*n + 48)*x - 54*x^2 - (81*n^2 - 324*n + 252)*x^3)*deriv(v[i - 4], x) + (54*n^3 - 351*n^2 + 663*n - 336 + (27*n^3 - 99*n^2 - 114*n + 586)*x - (27*n^2 + 90*n - 480)*x^2 - (81*n^3 - 540*n^2 + 1116*n - 528)*x^3)*v[i - 3] + (63*n^2 - 189*n + 112 + (27*n^3 - 234*n^2 + 507*n - 398)*x - (54*n^2 - 135*n - 174)*x^2 - (81*n^3 - 621*n^2 + 1440*n - 978)*x^3 + (81*n^2 - 324*n + 252)*x^4)*deriv(v[i - 3], x) + ((27*n^2 - 81*n + 48)*x - (27*n^2 - 81*n + 102)*x^2 - (81*n^2 - 324*n + 198)*x^3 + (81*n^2 - 324*n + 252)*x^4)*deriv(deriv(v[i - 3], x), x) - (9*n^2 + 45*n - 48 + (108*n^2 - 582*n + 454)*x + (45*n^2 - 240*n + 344)*x^2 + (144*n - 240)*x^3)*v[i - 2] - (9*n^2 - 27*n + 16 + (54*n - 234)*x - (27*n^2 - 54*n - 240)*x^2 + 36*x^3 - 144*x^4)*deriv(v[i - 2], x) + (72*x^2 - 144*x^3 + 72*x^4)*deriv(deriv(v[i - 2], x), x) + (18*n + 16 + (48*n - 164)*x + 48*x^2)*v[i - 1])/(8)); 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]
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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
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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 ›
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from
Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
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.