A144443 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(961 - 1860\,n + 900\,n^{2}\right)\,P_{n}(x) = \left(90396\,x^{2} - 311040\,n x^{2} + 291600\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(44640\,x^{3} - 138240\,n x^{3} + 129600\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(133176\,x - 273420\,x^{2} + 23040\,n x + 1316520\,n x^{2} - 21600\,n^{2} x - 1756080\,n^{2} x^{2} + 583200\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(40920\,x^{2} - 273048\,x^{3} - 23040\,n x^{2} + 1046304\,n x^{3} + 21600\,n^{2} x^{2} - 1205280\,n^{2} x^{3} + 259200\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(44640\,x^{3} - 44640\,x^{4} - 138240\,n x^{3} + 138240\,n x^{4} + 129600\,n^{2} x^{3} - 129600\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(155 - 221092\,x + 31713\,x^{2} - 4800\,n + 46296\,n x - 649512\,n x^{2} + 4500\,n^{2} + 84240\,n^{2} x + 1479384\,n^{2} x^{2} - 21600\,n^{3} x - 1153440\,n^{3} x^{2} + 291600\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(38688\,x - 193316\,x^{2} + 49972\,x^{3} + 11520\,n x - 67176\,n x^{2} - 811704\,n x^{3} - 10800\,n^{2} x + 66960\,n^{2} x^{2} + 1569024\,n^{2} x^{3} + 21600\,n^{3} x^{2} - 937440\,n^{3} x^{3} + 129600\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(32364\,x^{2} - 43896\,x^{3} + 11532\,x^{4} + 34560\,n x^{2} + 126144\,n x^{3} - 160704\,n x^{4} - 32400\,n^{2} x^{2} - 246240\,n^{2} x^{3} + 278640\,n^{2} x^{4} + 129600\,n^{3} x^{3} - 129600\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(806 - 19220\,x + 2940\,n + 60264\,n x - 3600\,n^{2} - 62640\,n^{2} x + 21600\,n^{3} x\right)\,P_{n-1}(x) + \left(372\,x - 3844\,x^{2} - 16920\,n x - 15624\,n x^{2} + 10800\,n^{2} x + 41040\,n^{2} x^{2} - 21600\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
\(P_{2}(x) = 1 + 80\,x + 9\,x^{2}\)
\(P_{3}(x) = 1 + 636\,x + 1431\,x^{2} + 16\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 1 | 80 | 9 | 90 | |||||
| 3 | 1 | 636 | 1431 | 16 | 2084 | ||||
| 4 | 1 | 4552 | 38286 | 18208 | 25 | 61072 | |||
| 5 | 1 | 31988 | 700902 | 1246048 | 199925 | 36 | 2178900 | ||
| 6 | 1 | 224064 | 10972503 | 47758208 | 30479175 | 2016576 | 49 | 91450576 | |
| 7 | 1 | 1568620 | 158702841 | 1402549040 | 2191482875 | 634811364 | 19215595 | 64 | 4408330400 |
Compute more
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Technical details
- Node id
n-4aa6c410d6c94a43· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-b8d0c092bbd5a0ce
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, 80, 9; 3: 1, 636, 1431, 16; 4: 1, 4552, 38286, 18208, 25; 5: 1, 31988, 700902, 1246048, 199925, 36; 6: 1, 224064, 10972503, 47758208, 30479175, 2016576, 49; 7: 1, 1568620, 158702841, 1402549040, 2191482875, 634811364, 19215595, 64; 8: 1, 10980536, 2196749412, 35511761696, 110021876950, 79901463816, 11960080132, 175688576, 81; 9: 1, 76863972, 29641285584, 820913597184, 4483796778150, 6456667360536, 2514047891376, 210782475264, 1556495433, 100;
%F FormulaThe row polynomials satisfy (900*n^2-1860*n+961)*P(n,x) = (291600*n^2-311040*n+90396)*x^2*P(n-4,x) + (129600*n^2-138240*n+44640)*x^3*d/dx P(n-4,x) - ((21600*n^2-23040*n-133176)*x-(583200*n^3-1756080*n^2+1316520*n-273420)*x^2)*P(n-3,x) + ((21600*n^2-23040*n+40920)*x^2+(259200*n^3-1205280*n^2+1046304*n-273048)*x^3)*d/dx P(n-3,x) + ((129600*n^2-138240*n+44640)*x^3-(129600*n^2-138240*n+44640)*x^4)*d^2/dx^2 P(n-3,x) + (4500*n^2-4800*n+155-(21600*n^3-84240*n^2-46296*n+221092)*x+(291600*n^4-1153440*n^3+1479384*n^2-649512*n+31713)*x^2)*P(n-2,x) - ((10800*n^2-11520*n-38688)*x-(21600*n^3+66960*n^2-67176*n-193316)*x^2-(129600*n^4-937440*n^3+1569024*n^2-811704*n+49972)*x^3)*d/dx P(n-2,x) - ((32400*n^2-34560*n-32364)*x^2-(129600*n^3-246240*n^2+126144*n-43896)*x^3+(129600*n^3-278640*n^2+160704*n-11532)*x^4)*d^2/dx^2 P(n-2,x) - (3600*n^2-2940*n-806-(21600*n^3-62640*n^2+60264*n-19220)*x)*P(n-1,x) + ((10800*n^2-16920*n+372)*x-(21600*n^3-41040*n^2+15624*n+3844)*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 + 80*x + 9*x^2, P(3,x) = 1 + 636*x + 1431*x^2 + 16*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 80*x + 9*x^2;
P[3, x_] := 1 + 636*x + 1431*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((291600*n^2 - 311040*n + 90396)*x^2*P[n - 4, x] + (129600*n^2 - 138240*n + 44640)*x^3*D[P[n - 4, x], x] - ((21600*n^2 - 23040*n - 133176)*x - (583200*n^3 - 1756080*n^2 + 1316520*n - 273420)*x^2)*P[n - 3, x] + ((21600*n^2 - 23040*n + 40920)*x^2 + (259200*n^3 - 1205280*n^2 + 1046304*n - 273048)*x^3)*D[P[n - 3, x], x] + ((129600*n^2 - 138240*n + 44640)*x^3 - (129600*n^2 - 138240*n + 44640)*x^4)*D[P[n - 3, x], {x, 2}] + (4500*n^2 - 4800*n + 155 - (21600*n^3 - 84240*n^2 - 46296*n + 221092)*x + (291600*n^4 - 1153440*n^3 + 1479384*n^2 - 649512*n + 31713)*x^2)*P[n - 2, x] - ((10800*n^2 - 11520*n - 38688)*x - (21600*n^3 + 66960*n^2 - 67176*n - 193316)*x^2 - (129600*n^4 - 937440*n^3 + 1569024*n^2 - 811704*n + 49972)*x^3)*D[P[n - 2, x], x] - ((32400*n^2 - 34560*n - 32364)*x^2 - (129600*n^3 - 246240*n^2 + 126144*n - 43896)*x^3 + (129600*n^3 - 278640*n^2 + 160704*n - 11532)*x^4)*D[P[n - 2, x], {x, 2}] - (3600*n^2 - 2940*n - 806 - (21600*n^3 - 62640*n^2 + 60264*n - 19220)*x)*P[n - 1, x] + ((10800*n^2 - 16920*n + 372)*x - (21600*n^3 - 41040*n^2 + 15624*n + 3844)*x^2)*D[P[n - 1, x], x])/(900*n^2 - 1860*n + 961)]];
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 + 80*x + 9*x^2; v[4] = 1 + 636*x + 1431*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((291600*n^2 - 311040*n + 90396)*x^2*v[i - 4] + (129600*n^2 - 138240*n + 44640)*x^3*deriv(v[i - 4], x) - ((21600*n^2 - 23040*n - 133176)*x - (583200*n^3 - 1756080*n^2 + 1316520*n - 273420)*x^2)*v[i - 3] + ((21600*n^2 - 23040*n + 40920)*x^2 + (259200*n^3 - 1205280*n^2 + 1046304*n - 273048)*x^3)*deriv(v[i - 3], x) + ((129600*n^2 - 138240*n + 44640)*x^3 - (129600*n^2 - 138240*n + 44640)*x^4)*deriv(deriv(v[i - 3], x), x) + (4500*n^2 - 4800*n + 155 - (21600*n^3 - 84240*n^2 - 46296*n + 221092)*x + (291600*n^4 - 1153440*n^3 + 1479384*n^2 - 649512*n + 31713)*x^2)*v[i - 2] - ((10800*n^2 - 11520*n - 38688)*x - (21600*n^3 + 66960*n^2 - 67176*n - 193316)*x^2 - (129600*n^4 - 937440*n^3 + 1569024*n^2 - 811704*n + 49972)*x^3)*deriv(v[i - 2], x) - ((32400*n^2 - 34560*n - 32364)*x^2 - (129600*n^3 - 246240*n^2 + 126144*n - 43896)*x^3 + (129600*n^3 - 278640*n^2 + 160704*n - 11532)*x^4)*deriv(deriv(v[i - 2], x), x) - (3600*n^2 - 2940*n - 806 - (21600*n^3 - 62640*n^2 + 60264*n - 19220)*x)*v[i - 1] + ((10800*n^2 - 16920*n + 372)*x - (21600*n^3 - 41040*n^2 + 15624*n + 3844)*x^2)*deriv(v[i - 1], x))/(900*n^2 - 1860*n + 961)); 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) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*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) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*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.