A144442 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(40\right)\,P_{n}(x) = \left(1780 - 610\,x - 1750\,x^{2} - 7200\,x^{3} - 2450\,n - 1225\,n x + 8000\,n x^{3} + 750\,n^{2} + 375\,n^{2} x - 1875\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(890\,x - 750\,x^{2} - 7200\,x^{3} - 1225\,n x + 8000\,n x^{3} + 375\,n^{2} x - 1875\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-6586 + 8638\,x + 6760\,x^{2} + 18160\,x^{3} + 10845\,n - 1130\,n x - 1150\,n x^{2} - 29600\,n x^{3} - 5225\,n^{2} - 1525\,n^{2} x - 375\,n^{2} x^{2} + 13250\,n^{2} x^{3} + 750\,n^{3} + 375\,n^{3} x - 1875\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(1958 - 6812\,x + 2020\,x^{2} + 28110\,x^{3} + 7200\,x^{4} - 2695\,n + 7995\,n x + 2075\,n x^{2} - 37600\,n x^{3} - 8000\,n x^{4} + 825\,n^{2} - 3400\,n^{2} x - 750\,n^{2} x^{2} + 15125\,n^{2} x^{3} + 1875\,n^{2} x^{4} + 375\,n^{3} x - 1875\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(890\,x - 1640\,x^{2} - 6450\,x^{3} + 7200\,x^{4} - 1225\,n x + 1225\,n x^{2} + 8000\,n x^{3} - 8000\,n x^{4} + 375\,n^{2} x - 375\,n^{2} x^{2} - 1875\,n^{2} x^{3} + 1875\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(902 - 8990\,x - 4900\,x^{2} + 3600\,x^{3} - 655\,n + 8750\,n x + 3500\,n x^{2} - 2000\,n x^{3} - 75\,n^{2} - 1500\,n^{2} x - 625\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-178 + 3250\,x - 3160\,x^{2} - 500\,x^{3} + 2000\,x^{4} + 245\,n - 750\,n x - 850\,n x^{2} - 75\,n^{2} + 375\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(1000\,x^{2} - 2000\,x^{3} + 1000\,x^{4}\right)\,P_{n-2}''(x) + \left(60 - 1420\,x + 400\,x^{2} + 150\,n + 400\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 18 + 19\,x + x^{2}\)
\(P_{3}(x) = 119 + 354\,x + 121\,x^{2} + x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 18 | 19 | 1 | 38 | |||||
| 3 | 119 | 354 | 121 | 1 | 595 | ||||
| 4 | 730 | 6091 | 4868 | 733 | 1 | 12423 | |||
| 5 | 4401 | 87890 | 166980 | 59345 | 4405 | 1 | 323022 | ||
| 6 | 26432 | 1136911 | 4602975 | 3570205 | 687395 | 26437 | 1 | 10050356 | |
| 7 | 158623 | 13792314 | 107186901 | 167283425 | 67540915 | 7768578 | 158629 | 1 | 363889386 |
Compute more
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Technical details
- Node id
n-eca254ae7d32dfb0· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-e76b964912e6b0dd
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: 18, 19, 1; 3: 119, 354, 121, 1; 4: 730, 6091, 4868, 733, 1; 5: 4401, 87890, 166980, 59345, 4405, 1; 6: 26432, 1136911, 4602975, 3570205, 687395, 26437, 1; 7: 158623, 13792314, 107186901, 167283425, 67540915, 7768578, 158629, 1; 8: 951774, 161021259, 2225998211, 6363029136, 4988532775, 1195529614, 86697154, 951781, 1; 9: 5710685, 1836588906, 42796338388, 207624701263, 290496602256, 132177898930, 20368373536, 961124583, 5710693, 1;
%F FormulaThe row polynomials satisfy 40*P(n,x) = (750*n^2-2450*n+1780+(375*n^2-1225*n-610)*x-1750*x^2-(1875*n^2-8000*n+7200)*x^3)*P(n-4,x) + ((375*n^2-1225*n+890)*x-750*x^2-(1875*n^2-8000*n+7200)*x^3)*d/dx P(n-4,x) + (750*n^3-5225*n^2+10845*n-6586+(375*n^3-1525*n^2-1130*n+8638)*x-(375*n^2+1150*n-6760)*x^2-(1875*n^3-13250*n^2+29600*n-18160)*x^3)*P(n-3,x) + (825*n^2-2695*n+1958+(375*n^3-3400*n^2+7995*n-6812)*x-(750*n^2-2075*n-2020)*x^2-(1875*n^3-15125*n^2+37600*n-28110)*x^3+(1875*n^2-8000*n+7200)*x^4)*d/dx P(n-3,x) + ((375*n^2-1225*n+890)*x-(375*n^2-1225*n+1640)*x^2-(1875*n^2-8000*n+6450)*x^3+(1875*n^2-8000*n+7200)*x^4)*d^2/dx^2 P(n-3,x) - (75*n^2+655*n-902+(1500*n^2-8750*n+8990)*x+(625*n^2-3500*n+4900)*x^2+(2000*n-3600)*x^3)*P(n-2,x) - (75*n^2-245*n+178+(750*n-3250)*x-(375*n^2-850*n-3160)*x^2+500*x^3-2000*x^4)*d/dx P(n-2,x) + (1000*x^2-2000*x^3+1000*x^4)*d^2/dx^2 P(n-2,x) + (150*n+60+(400*n-1420)*x+400*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 18 + 19*x + x^2, P(3,x) = 119 + 354*x + 121*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 18 + 19*x + x^2;
P[3, x_] := 119 + 354*x + 121*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((750*n^2 - 2450*n + 1780 + (375*n^2 - 1225*n - 610)*x - 1750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*P[n - 4, x] + ((375*n^2 - 1225*n + 890)*x - 750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*D[P[n - 4, x], x] + (750*n^3 - 5225*n^2 + 10845*n - 6586 + (375*n^3 - 1525*n^2 - 1130*n + 8638)*x - (375*n^2 + 1150*n - 6760)*x^2 - (1875*n^3 - 13250*n^2 + 29600*n - 18160)*x^3)*P[n - 3, x] + (825*n^2 - 2695*n + 1958 + (375*n^3 - 3400*n^2 + 7995*n - 6812)*x - (750*n^2 - 2075*n - 2020)*x^2 - (1875*n^3 - 15125*n^2 + 37600*n - 28110)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*D[P[n - 3, x], x] + ((375*n^2 - 1225*n + 890)*x - (375*n^2 - 1225*n + 1640)*x^2 - (1875*n^2 - 8000*n + 6450)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*D[P[n - 3, x], {x, 2}] - (75*n^2 + 655*n - 902 + (1500*n^2 - 8750*n + 8990)*x + (625*n^2 - 3500*n + 4900)*x^2 + (2000*n - 3600)*x^3)*P[n - 2, x] - (75*n^2 - 245*n + 178 + (750*n - 3250)*x - (375*n^2 - 850*n - 3160)*x^2 + 500*x^3 - 2000*x^4)*D[P[n - 2, x], x] + (1000*x^2 - 2000*x^3 + 1000*x^4)*D[P[n - 2, x], {x, 2}] + (150*n + 60 + (400*n - 1420)*x + 400*x^2)*P[n - 1, x])/(40)]];
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] = 18 + 19*x + x^2; v[4] = 119 + 354*x + 121*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((750*n^2 - 2450*n + 1780 + (375*n^2 - 1225*n - 610)*x - 1750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*v[i - 4] + ((375*n^2 - 1225*n + 890)*x - 750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*deriv(v[i - 4], x) + (750*n^3 - 5225*n^2 + 10845*n - 6586 + (375*n^3 - 1525*n^2 - 1130*n + 8638)*x - (375*n^2 + 1150*n - 6760)*x^2 - (1875*n^3 - 13250*n^2 + 29600*n - 18160)*x^3)*v[i - 3] + (825*n^2 - 2695*n + 1958 + (375*n^3 - 3400*n^2 + 7995*n - 6812)*x - (750*n^2 - 2075*n - 2020)*x^2 - (1875*n^3 - 15125*n^2 + 37600*n - 28110)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*deriv(v[i - 3], x) + ((375*n^2 - 1225*n + 890)*x - (375*n^2 - 1225*n + 1640)*x^2 - (1875*n^2 - 8000*n + 6450)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*deriv(deriv(v[i - 3], x), x) - (75*n^2 + 655*n - 902 + (1500*n^2 - 8750*n + 8990)*x + (625*n^2 - 3500*n + 4900)*x^2 + (2000*n - 3600)*x^3)*v[i - 2] - (75*n^2 - 245*n + 178 + (750*n - 3250)*x - (375*n^2 - 850*n - 3160)*x^2 + 500*x^3 - 2000*x^4)*deriv(v[i - 2], x) + (1000*x^2 - 2000*x^3 + 1000*x^4)*deriv(deriv(v[i - 2], x), x) + (150*n + 60 + (400*n - 1420)*x + 400*x^2)*v[i - 1])/(40)); 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 read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
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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 read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
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.