A144442 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(441 - 840\,n + 400\,n^{2}\right)\,P_{n}(x) = \left(30975\,x^{2} - 99000\,n x^{2} + 90000\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(15225\,x^{3} - 44000\,n x^{3} + 40000\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(41580\,x - 88935\,x^{2} + 8800\,n x + 411600\,n x^{2} - 8000\,n^{2} x - 542000\,n^{2} x^{2} + 180000\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(13545\,x^{2} - 91035\,x^{3} - 8800\,n x^{2} + 329725\,n x^{3} + 8000\,n^{2} x^{2} - 372000\,n^{2} x^{3} + 80000\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(15225\,x^{3} - 15225\,x^{4} - 44000\,n x^{3} + 44000\,n x^{4} + 40000\,n^{2} x^{3} - 40000\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(84 - 66024\,x + 3969\,x^{2} - 1760\,n + 8940\,n x - 188370\,n x^{2} + 1600\,n^{2} + 31200\,n^{2} x + 447225\,n^{2} x^{2} - 8000\,n^{3} x - 353000\,n^{3} x^{2} + 90000\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(12390\,x - 59556\,x^{2} + 8379\,x^{3} + 4400\,n x - 19740\,n x^{2} - 240870\,n x^{3} - 4000\,n^{2} x + 16800\,n^{2} x^{2} + 480100\,n^{2} x^{3} + 8000\,n^{3} x^{2} - 288000\,n^{3} x^{3} + 40000\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(9975\,x^{2} - 12180\,x^{3} + 2205\,x^{4} + 11000\,n x^{2} + 37300\,n x^{3} - 48300\,n x^{4} - 10000\,n^{2} x^{2} - 76000\,n^{2} x^{3} + 86000\,n^{2} x^{4} + 40000\,n^{3} x^{3} - 40000\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(357 - 7056\,x + 920\,n + 22260\,n x - 1200\,n^{2} - 23200\,n^{2} x + 8000\,n^{3} x\right)\,P_{n-1}(x) + \left(210\,x - 1764\,x^{2} - 6400\,n x - 5460\,n x^{2} + 4000\,n^{2} x + 15200\,n^{2} x^{2} - 8000\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
\(P_{2}(x) = 1 + 68\,x + 9\,x^{2}\)
\(P_{3}(x) = 1 + 472\,x + 1062\,x^{2} + 16\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 1 | 68 | 9 | 78 | |||||
| 3 | 1 | 472 | 1062 | 16 | 1551 | ||||
| 4 | 1 | 2916 | 24129 | 11664 | 25 | 38735 | |||
| 5 | 1 | 17600 | 375705 | 667920 | 110000 | 36 | 1171262 | ||
| 6 | 1 | 105724 | 4997160 | 21587920 | 13881000 | 951516 | 49 | 41523370 | |
| 7 | 1 | 634488 | 61351614 | 535306960 | 836417125 | 245406456 | 7772478 | 64 | 1686889186 |
Compute more
?
Technical details
- Node id
n-86c57fc08f195991· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-c5e5ab6ef10668e5
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: 1, 4; 2: 1, 68, 9; 3: 1, 472, 1062, 16; 4: 1, 2916, 24129, 11664, 25; 5: 1, 17600, 375705, 667920, 110000, 36; 6: 1, 105724, 4997160, 21587920, 13881000, 951516, 49; 7: 1, 634488, 61351614, 535306960, 836417125, 245406456, 7772478, 64; 8: 1, 3807092, 720312687, 11445138496, 35298174875, 25751561616, 3921702407, 60913472, 81; 9: 1, 22842736, 8238951999, 223364796144, 1210402509400, 1742979613536, 684054688191, 58588103104, 462565404, 100;
%F FormulaThe row polynomials satisfy (400*n^2-840*n+441)*P(n,x) = (90000*n^2-99000*n+30975)*x^2*P(n-4,x) + (40000*n^2-44000*n+15225)*x^3*d/dx P(n-4,x) - ((8000*n^2-8800*n-41580)*x-(180000*n^3-542000*n^2+411600*n-88935)*x^2)*P(n-3,x) + ((8000*n^2-8800*n+13545)*x^2+(80000*n^3-372000*n^2+329725*n-91035)*x^3)*d/dx P(n-3,x) + ((40000*n^2-44000*n+15225)*x^3-(40000*n^2-44000*n+15225)*x^4)*d^2/dx^2 P(n-3,x) + (1600*n^2-1760*n+84-(8000*n^3-31200*n^2-8940*n+66024)*x+(90000*n^4-353000*n^3+447225*n^2-188370*n+3969)*x^2)*P(n-2,x) - ((4000*n^2-4400*n-12390)*x-(8000*n^3+16800*n^2-19740*n-59556)*x^2-(40000*n^4-288000*n^3+480100*n^2-240870*n+8379)*x^3)*d/dx P(n-2,x) - ((10000*n^2-11000*n-9975)*x^2-(40000*n^3-76000*n^2+37300*n-12180)*x^3+(40000*n^3-86000*n^2+48300*n-2205)*x^4)*d^2/dx^2 P(n-2,x) - (1200*n^2-920*n-357-(8000*n^3-23200*n^2+22260*n-7056)*x)*P(n-1,x) + ((4000*n^2-6400*n+210)*x-(8000*n^3-15200*n^2+5460*n+1764)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 68*x + 9*x^2, P(3,x) = 1 + 472*x + 1062*x^2 + 16*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 68*x + 9*x^2;
P[3, x_] := 1 + 472*x + 1062*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((90000*n^2 - 99000*n + 30975)*x^2*P[n - 4, x] + (40000*n^2 - 44000*n + 15225)*x^3*D[P[n - 4, x], x] - ((8000*n^2 - 8800*n - 41580)*x - (180000*n^3 - 542000*n^2 + 411600*n - 88935)*x^2)*P[n - 3, x] + ((8000*n^2 - 8800*n + 13545)*x^2 + (80000*n^3 - 372000*n^2 + 329725*n - 91035)*x^3)*D[P[n - 3, x], x] + ((40000*n^2 - 44000*n + 15225)*x^3 - (40000*n^2 - 44000*n + 15225)*x^4)*D[P[n - 3, x], {x, 2}] + (1600*n^2 - 1760*n + 84 - (8000*n^3 - 31200*n^2 - 8940*n + 66024)*x + (90000*n^4 - 353000*n^3 + 447225*n^2 - 188370*n + 3969)*x^2)*P[n - 2, x] - ((4000*n^2 - 4400*n - 12390)*x - (8000*n^3 + 16800*n^2 - 19740*n - 59556)*x^2 - (40000*n^4 - 288000*n^3 + 480100*n^2 - 240870*n + 8379)*x^3)*D[P[n - 2, x], x] - ((10000*n^2 - 11000*n - 9975)*x^2 - (40000*n^3 - 76000*n^2 + 37300*n - 12180)*x^3 + (40000*n^3 - 86000*n^2 + 48300*n - 2205)*x^4)*D[P[n - 2, x], {x, 2}] - (1200*n^2 - 920*n - 357 - (8000*n^3 - 23200*n^2 + 22260*n - 7056)*x)*P[n - 1, x] + ((4000*n^2 - 6400*n + 210)*x - (8000*n^3 - 15200*n^2 + 5460*n + 1764)*x^2)*D[P[n - 1, x], x])/(400*n^2 - 840*n + 441)]];
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 + 4*x; v[3] = 1 + 68*x + 9*x^2; v[4] = 1 + 472*x + 1062*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((90000*n^2 - 99000*n + 30975)*x^2*v[i - 4] + (40000*n^2 - 44000*n + 15225)*x^3*deriv(v[i - 4], x) - ((8000*n^2 - 8800*n - 41580)*x - (180000*n^3 - 542000*n^2 + 411600*n - 88935)*x^2)*v[i - 3] + ((8000*n^2 - 8800*n + 13545)*x^2 + (80000*n^3 - 372000*n^2 + 329725*n - 91035)*x^3)*deriv(v[i - 3], x) + ((40000*n^2 - 44000*n + 15225)*x^3 - (40000*n^2 - 44000*n + 15225)*x^4)*deriv(deriv(v[i - 3], x), x) + (1600*n^2 - 1760*n + 84 - (8000*n^3 - 31200*n^2 - 8940*n + 66024)*x + (90000*n^4 - 353000*n^3 + 447225*n^2 - 188370*n + 3969)*x^2)*v[i - 2] - ((4000*n^2 - 4400*n - 12390)*x - (8000*n^3 + 16800*n^2 - 19740*n - 59556)*x^2 - (40000*n^4 - 288000*n^3 + 480100*n^2 - 240870*n + 8379)*x^3)*deriv(v[i - 2], x) - ((10000*n^2 - 11000*n - 9975)*x^2 - (40000*n^3 - 76000*n^2 + 37300*n - 12180)*x^3 + (40000*n^3 - 86000*n^2 + 48300*n - 2205)*x^4)*deriv(deriv(v[i - 2], x), x) - (1200*n^2 - 920*n - 357 - (8000*n^3 - 23200*n^2 + 22260*n - 7056)*x)*v[i - 1] + ((4000*n^2 - 6400*n + 210)*x - (8000*n^3 - 15200*n^2 + 5460*n + 1764)*x^2)*deriv(v[i - 1], x))/(400*n^2 - 840*n + 441)); 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 multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, 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 ≥ 1
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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
- [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.