A257626 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(-36 - 54\,n - 180\,n x + 144\,n x^{2} - 18\,n^{2} - 216\,n^{2} x + 369\,n^{2} x^{2} - 72\,n^{3} x + 306\,n^{3} x^{2} + 81\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-99\,x + 270\,x^{2} - 207\,x^{3} - 108\,n x + 477\,n x^{2} - 468\,n x^{3} - 36\,n^{2} x + 324\,n^{2} x^{2} - 279\,n^{2} x^{3} + 72\,n^{3} x^{2} + 18\,n^{3} x^{3} + 36\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-27\,x^{2} + 90\,x^{3} - 63\,x^{4} - 27\,n x^{2} + 189\,n x^{3} - 162\,n x^{4} - 9\,n^{2} x^{2} + 144\,n^{2} x^{3} - 135\,n^{2} x^{4} + 36\,n^{3} x^{3} - 36\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(12 + 12\,x + 21\,n + 36\,n x + 9\,n^{2} + 36\,n^{2} x + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 24\,x^{2} + 15\,n x - 60\,n x^{2} + 6\,n^{2} x - 48\,n^{2} x^{2} - 12\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 24\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 6 | 24 | 30 | ||||||
| 2 | 36 | 432 | 324 | 792 | |||||
| 3 | 216 | 5616 | 12636 | 3456 | 21924 | ||||
| 4 | 1296 | 63504 | 303264 | 254016 | 32400 | 654480 | |||
| 5 | 7776 | 664848 | 5782428 | 10279872 | 4155300 | 279936 | 21170160 | ||
| 6 | 46656 | 6636816 | 96315480 | 308396160 | 267543000 | 59731344 | 2286144 | 740955600 | |
| 7 | 279936 | 64210320 | 1469375316 | 7708037760 | 12043809000 | 5877501264 | 786576420 | 17915904 | 27967705920 |
Compute more
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Technical details
- Node id
n-dd4ba6fa28007a1f· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-5cf267279c5da1ec
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: 6, 24; 2: 36, 432, 324; 3: 216, 5616, 12636, 3456; 4: 1296, 63504, 303264, 254016, 32400; 5: 7776, 664848, 5782428, 10279872, 4155300, 279936; 6: 46656, 6636816, 96315480, 308396160, 267543000, 59731344, 2286144; 7: 279936, 64210320, 1469375316, 7708037760, 12043809000, 5877501264, 786576420, 17915904; 8: 1679616, 608125968, 21099861072, 170477244864, 433577124000, 383573800944, 114877021392, 9730015488, 136048896; 9: 10077696, 5674687632, 290141985420, 3457419412032, 13398172829100, 19293368873904, 10588346949348, 2063231896320, 114912424548, 1007769600;
%F FormulaThe row polynomials satisfy (n^2+2*n+1)*P(n,x) = -(18*n^2+54*n+36+(72*n^3+216*n^2+180*n)*x-(81*n^4+306*n^3+369*n^2+144*n)*x^2)*P(n-2,x) - ((36*n^2+108*n+99)*x-(72*n^3+324*n^2+477*n+270)*x^2-(36*n^4+18*n^3-279*n^2-468*n-207)*x^3)*d/dx P(n-2,x) - ((9*n^2+27*n+27)*x^2-(36*n^3+144*n^2+189*n+90)*x^3+(36*n^3+135*n^2+162*n+63)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2+21*n+12+(12*n^3+36*n^2+36*n+12)*x)*P(n-1,x) + ((6*n^2+15*n+12)*x-(12*n^3+48*n^2+60*n+24)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 24*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 6 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 + 54*n + 36 + (72*n^3 + 216*n^2 + 180*n)*x - (81*n^4 + 306*n^3 + 369*n^2 + 144*n)*x^2)*P[n - 2, x] - ((36*n^2 + 108*n + 99)*x - (72*n^3 + 324*n^2 + 477*n + 270)*x^2 - (36*n^4 + 18*n^3 - 279*n^2 - 468*n - 207)*x^3)*D[P[n - 2, x], x] - ((9*n^2 + 27*n + 27)*x^2 - (36*n^3 + 144*n^2 + 189*n + 90)*x^3 + (36*n^3 + 135*n^2 + 162*n + 63)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 + 21*n + 12 + (12*n^3 + 36*n^2 + 36*n + 12)*x)*P[n - 1, x] + ((6*n^2 + 15*n + 12)*x - (12*n^3 + 48*n^2 + 60*n + 24)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 + 54*n + 36 + (72*n^3 + 216*n^2 + 180*n)*x - (81*n^4 + 306*n^3 + 369*n^2 + 144*n)*x^2)*v[i - 2] - ((36*n^2 + 108*n + 99)*x - (72*n^3 + 324*n^2 + 477*n + 270)*x^2 - (36*n^4 + 18*n^3 - 279*n^2 - 468*n - 207)*x^3)*deriv(v[i - 2], x) - ((9*n^2 + 27*n + 27)*x^2 - (36*n^3 + 144*n^2 + 189*n + 90)*x^3 + (36*n^3 + 135*n^2 + 162*n + 63)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 + 21*n + 12 + (12*n^3 + 36*n^2 + 36*n + 12)*x)*v[i - 1] + ((6*n^2 + 15*n + 12)*x - (12*n^3 + 48*n^2 + 60*n + 24)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n + 1)); 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]
-
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
-
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
Related ?
Same fingerprint ?
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.