A144441 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-13 + 12\,n\right)\,P_{n}(x) = \left(16\,x^{2} - 192\,n x^{2}\right)\,P_{n-4}(x) + \left(-24\,x + 8\,x^{2} + 288\,n x + 704\,n x^{2} - 384\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(16\,x^{2} - 16\,x^{3} - 192\,n x^{2} + 192\,n x^{3}\right)\,P_{n-3}'(x) + \left(-3 - 62\,x - 91\,x^{2} + 36\,n - 432\,n x - 228\,n x^{2} + 288\,n^{2} x + 496\,n^{2} x^{2} - 192\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 64\,x^{2} - 52\,x^{3} + 144\,n x + 16\,n x^{2} - 160\,n x^{3} - 192\,n^{2} x^{2} + 192\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-10 + 78\,x - 24\,n - 176\,n x + 96\,n^{2} x\right)\,P_{n-1}(x) + \left(-52\,x + 52\,x^{2} + 48\,n x - 48\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 28\,x + 3\,x^{2}\)
\(P_{3}(x) = 1 + 166\,x + 249\,x^{2} + 4\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 28 | 3 | 32 | |||||
| 3 | 1 | 166 | 249 | 4 | 420 | ||||
| 4 | 1 | 864 | 4650 | 1728 | 5 | 7248 | |||
| 5 | 1 | 4362 | 59694 | 79592 | 10905 | 6 | 154560 | ||
| 6 | 1 | 21860 | 653661 | 2094192 | 1089435 | 65580 | 7 | 3924736 | |
| 7 | 1 | 109358 | 6597711 | 42359180 | 52948975 | 13195422 | 382753 | 8 | 115593408 |
Compute more
?
Technical details
- Node id
n-62d88627e028c40c· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-d85ced95fe65dfa0
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, 2; 2: 1, 28, 3; 3: 1, 166, 249, 4; 4: 1, 864, 4650, 1728, 5; 5: 1, 4362, 59694, 79592, 10905, 6; 6: 1, 21860, 653661, 2094192, 1089435, 65580, 7; 7: 1, 109358, 6597711, 42359180, 52948975, 13195422, 382753, 8; 8: 1, 546856, 63611484, 738891440, 1810736110, 1108337160, 148426796, 2187424, 9; 9: 1, 2734354, 596947740, 11761159312, 50390210070, 60468252084, 20582028796, 1591860640, 12304593, 10;
%F FormulaThe row polynomials satisfy (12*n-13)*P(n,x) = -(192*n-16)*x^2*P(n-4,x) + ((288*n-24)*x-(384*n^2-704*n-8)*x^2)*P(n-3,x) - ((192*n-16)*x^2-(192*n-16)*x^3)*d/dx P(n-3,x) + (36*n-3+(288*n^2-432*n-62)*x-(192*n^3-496*n^2+228*n+91)*x^2)*P(n-2,x) + ((144*n-12)*x-(192*n^2-16*n-64)*x^2+(192*n^2-160*n-52)*x^3)*d/dx P(n-2,x) - (24*n+10-(96*n^2-176*n+78)*x)*P(n-1,x) + ((48*n-52)*x-(48*n-52)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 28*x + 3*x^2, P(3,x) = 1 + 166*x + 249*x^2 + 4*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 28*x + 3*x^2;
P[3, x_] := 1 + 166*x + 249*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n - 16)*x^2*P[n - 4, x] + ((288*n - 24)*x - (384*n^2 - 704*n - 8)*x^2)*P[n - 3, x] - ((192*n - 16)*x^2 - (192*n - 16)*x^3)*D[P[n - 3, x], x] + (36*n - 3 + (288*n^2 - 432*n - 62)*x - (192*n^3 - 496*n^2 + 228*n + 91)*x^2)*P[n - 2, x] + ((144*n - 12)*x - (192*n^2 - 16*n - 64)*x^2 + (192*n^2 - 160*n - 52)*x^3)*D[P[n - 2, x], x] - (24*n + 10 - (96*n^2 - 176*n + 78)*x)*P[n - 1, x] + ((48*n - 52)*x - (48*n - 52)*x^2)*D[P[n - 1, x], x])/(12*n - 13)]];
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 + 2*x; v[3] = 1 + 28*x + 3*x^2; v[4] = 1 + 166*x + 249*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(192*n - 16)*x^2*v[i - 4] + ((288*n - 24)*x - (384*n^2 - 704*n - 8)*x^2)*v[i - 3] - ((192*n - 16)*x^2 - (192*n - 16)*x^3)*deriv(v[i - 3], x) + (36*n - 3 + (288*n^2 - 432*n - 62)*x - (192*n^3 - 496*n^2 + 228*n + 91)*x^2)*v[i - 2] + ((144*n - 12)*x - (192*n^2 - 16*n - 64)*x^2 + (192*n^2 - 160*n - 52)*x^3)*deriv(v[i - 2], x) - (24*n + 10 - (96*n^2 - 176*n + 78)*x)*v[i - 1] + ((48*n - 52)*x - (48*n - 52)*x^2)*deriv(v[i - 1], x))/(12*n - 13)); 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 T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(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) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…?
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.