A257619 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(49 - 126\,n + 81\,n^{2}\right)\,P_{n}(x) = \left(-196 - 38080\,x - 630\,n + 15660\,n x - 63504\,n x^{2} + 1134\,n^{2} + 13608\,n^{2} x + 199017\,n^{2} x^{2} - 5832\,n^{3} x - 196830\,n^{3} x^{2} + 59049\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6741\,x - 34370\,x^{2} + 1225\,x^{3} + 1620\,n x - 6507\,n x^{2} - 83412\,n x^{3} - 2916\,n^{2} x + 18468\,n^{2} x^{2} + 224289\,n^{2} x^{3} + 5832\,n^{3} x^{2} - 173502\,n^{3} x^{3} + 26244\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(6237\,x^{2} - 6678\,x^{3} + 441\,x^{4} + 3645\,n x^{2} + 13365\,n x^{3} - 17010\,n x^{4} - 6561\,n^{2} x^{2} - 34992\,n^{2} x^{3} + 41553\,n^{2} x^{4} + 26244\,n^{3} x^{3} - 26244\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(196 - 1372\,x + 63\,n + 5292\,n x - 405\,n^{2} - 6804\,n^{2} x + 2916\,n^{3} x\right)\,P_{n-1}(x) + \left(-252\,x - 392\,x^{2} - 1539\,n x - 756\,n x^{2} + 1458\,n^{2} x + 3888\,n^{2} x^{2} - 2916\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 8 | 10 | ||||||
| 2 | 4 | 176 | 36 | 216 | |||||
| 3 | 8 | 2256 | 5076 | 128 | 7468 | ||||
| 4 | 16 | 25744 | 203040 | 102976 | 400 | 332176 | |||
| 5 | 32 | 285616 | 5740596 | 10205504 | 1785100 | 1152 | 18018000 | ||
| 6 | 64 | 3147792 | 139232088 | 591919232 | 386755800 | 28330128 | 3136 | 1149388240 | |
| 7 | 128 | 34640048 | 3117520764 | 26571558784 | 41518060600 | 12470083056 | 424340588 | 8192 | 84136212160 |
Compute more
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Technical details
- Node id
n-ae0db26e09972dd3· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-3e3ec6af73020f4a
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, 8; 2: 4, 176, 36; 3: 8, 2256, 5076, 128; 4: 16, 25744, 203040, 102976, 400; 5: 32, 285616, 5740596, 10205504, 1785100, 1152; 6: 64, 3147792, 139232088, 591919232, 386755800, 28330128, 3136; 7: 128, 34640048, 3117520764, 26571558784, 41518060600, 12470083056, 424340588, 8192; 8: 256, 381073808, 66715061328, 1031061384128, 3155372605600, 2319888114288, 363226445008, 6097180928, 20736; 9: 512, 4191887664, 1390033270980, 36542635134144, 195622729409700, 281696730349968, 111911820098316, 9884681038080, 84885725196, 51200;
%F FormulaThe row polynomials satisfy (81*n^2-126*n+49)*P(n,x) = (1134*n^2-630*n-196-(5832*n^3-13608*n^2-15660*n+38080)*x+(59049*n^4-196830*n^3+199017*n^2-63504*n)*x^2)*P(n-2,x) - ((2916*n^2-1620*n-6741)*x-(5832*n^3+18468*n^2-6507*n-34370)*x^2-(26244*n^4-173502*n^3+224289*n^2-83412*n+1225)*x^3)*d/dx P(n-2,x) - ((6561*n^2-3645*n-6237)*x^2-(26244*n^3-34992*n^2+13365*n-6678)*x^3+(26244*n^3-41553*n^2+17010*n-441)*x^4)*d^2/dx^2 P(n-2,x) - (405*n^2-63*n-196-(2916*n^3-6804*n^2+5292*n-1372)*x)*P(n-1,x) + ((1458*n^2-1539*n-252)*x-(2916*n^3-3888*n^2+756*n+392)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((1134*n^2 - 630*n - 196 - (5832*n^3 - 13608*n^2 - 15660*n + 38080)*x + (59049*n^4 - 196830*n^3 + 199017*n^2 - 63504*n)*x^2)*P[n - 2, x] - ((2916*n^2 - 1620*n - 6741)*x - (5832*n^3 + 18468*n^2 - 6507*n - 34370)*x^2 - (26244*n^4 - 173502*n^3 + 224289*n^2 - 83412*n + 1225)*x^3)*D[P[n - 2, x], x] - ((6561*n^2 - 3645*n - 6237)*x^2 - (26244*n^3 - 34992*n^2 + 13365*n - 6678)*x^3 + (26244*n^3 - 41553*n^2 + 17010*n - 441)*x^4)*D[P[n - 2, x], {x, 2}] - (405*n^2 - 63*n - 196 - (2916*n^3 - 6804*n^2 + 5292*n - 1372)*x)*P[n - 1, x] + ((1458*n^2 - 1539*n - 252)*x - (2916*n^3 - 3888*n^2 + 756*n + 392)*x^2)*D[P[n - 1, x], x])/(81*n^2 - 126*n + 49)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((1134*n^2 - 630*n - 196 - (5832*n^3 - 13608*n^2 - 15660*n + 38080)*x + (59049*n^4 - 196830*n^3 + 199017*n^2 - 63504*n)*x^2)*v[i - 2] - ((2916*n^2 - 1620*n - 6741)*x - (5832*n^3 + 18468*n^2 - 6507*n - 34370)*x^2 - (26244*n^4 - 173502*n^3 + 224289*n^2 - 83412*n + 1225)*x^3)*deriv(v[i - 2], x) - ((6561*n^2 - 3645*n - 6237)*x^2 - (26244*n^3 - 34992*n^2 + 13365*n - 6678)*x^3 + (26244*n^3 - 41553*n^2 + 17010*n - 441)*x^4)*deriv(deriv(v[i - 2], x), x) - (405*n^2 - 63*n - 196 - (2916*n^3 - 6804*n^2 + 5292*n - 1372)*x)*v[i - 1] + ((1458*n^2 - 1539*n - 252)*x - (2916*n^3 - 3888*n^2 + 756*n + 392)*x^2)*deriv(v[i - 1], x))/(81*n^2 - 126*n + 49)); 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
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.