A142075 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-16\,x^{2} + 16\,n x + 96\,n x^{2} - 8\,n^{2} x - 116\,n^{2} x^{2} + 36\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x^{2} - 20\,x^{3} - 2\,n x + 108\,n x^{3} + 8\,n^{2} x^{2} - 104\,n^{2} x^{3} + 16\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x^{3} - 4\,x^{4} - n x^{2} - 8\,n x^{3} + 20\,n x^{4} + 8\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-1 + 8\,x + n - 16\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 2\,n x + 8\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 1 | 8 | 9 | |||||||||||||
| 3 | 1 | 32 | 36 | 69 | ||||||||||||
| 4 | 1 | 88 | 396 | 128 | 613 | |||||||||||
| 5 | 1 | 208 | 2376 | 3328 | 400 | 6313 | ||||||||||
| 6 | 1 | 456 | 10872 | 38656 | 22800 | 1152 | 73937 | |||||||||
| 7 | 1 | 960 | 42876 | 309248 | 476400 | 138240 | 3136 | 970861 | ||||||||
| 8 | 1 | 1976 | 154548 | 1999232 | 6247600 | 4945536 | 774592 | 8192 | 14131677 | |||||||
| 9 | 1 | 4016 | 525888 | 11293952 | 62476000 | 101645568 | 45810688 | 4112384 | 20736 | 225889233 | ||||||
| 10 | 1 | 8104 | 1722240 | 58264576 | 524141600 | 1509527808 | 1427482112 | 391905280 | 21005568 | 51200 | 3934108489 | |||||
| 11 | 1 | 16288 | 5494932 | 282046464 | 3895245600 | 18114333696 | 30538725504 | 18050973696 | 3165080832 | 104243200 | 123904 | 74156284117 | ||||
| 12 | 1 | 32664 | 17217756 | 1304023680 | 26527389600 | 187214153472 | 509638528896 | 543280939008 | 211251836160 | 24487475200 | 505900032 | 294912 | 1504227791381 | |||
| 13 | 1 | 65424 | 53270136 | 5828281600 | 169312614000 | 1734475977216 | 7134939404544 | 12334051393536 | 8777165909760 | 2331312640000 | 183343970304 | 2411790336 | 692224 | 32672896009081 | ||
| 14 | 1 | 130952 | 163343304 | 25396580608 | 1028696870000 | 14795438818176 | 87717608224512 | 229139874545664 | 266317898727168 | 131673199360000 | 24583890028544 | 1338108346368 | 11331014656 | 1605632 | 756631607595585 | |
| 15 | 1 | 262016 | 497690604 | 108555636736 | 6016491808400 | 118399102765056 | 976511508616896 | 3666237992730624 | 6456933240650496 | 5262182345113600 | 1863668502569984 | 250112187039744 | 9569816129536 | 52587659264 | 3686400 | 18609792832359357 |
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Technical details
- Node id
n-8ef7a22f007ce91e· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-c7274fd18fd89701
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 -1)
- 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1, 8; 3: 1, 32, 36; 4: 1, 88, 396, 128; 5: 1, 208, 2376, 3328, 400; 6: 1, 456, 10872, 38656, 22800, 1152; 7: 1, 960, 42876, 309248, 476400, 138240, 3136; 8: 1, 1976, 154548, 1999232, 6247600, 4945536, 774592, 8192; 9: 1, 4016, 525888, 11293952, 62476000, 101645568, 45810688, 4112384, 20736; 10: 1, 8104, 1722240, 58264576, 524141600, 1509527808, 1427482112, 391905280, 21005568, 51200;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = -((8*n^2-16*n)*x-(36*n^3-116*n^2+96*n-16)*x^2)*P(n-2,x) - (2*n*x-(8*n^2+2)*x^2-(16*n^3-104*n^2+108*n-20)*x^3)*d/dx P(n-2,x) - (n*x^2-(8*n^2-8*n+2)*x^3+(16*n^2-20*n+4)*x^4)*d^2/dx^2 P(n-2,x) + (n-1+(8*n^2-16*n+8)*x)*P(n-1,x) + ((2*n-1)*x-(8*n^2-8*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 - 16*n)*x - (36*n^3 - 116*n^2 + 96*n - 16)*x^2)*P[n - 2, x] - (2*n*x - (8*n^2 + 2)*x^2 - (16*n^3 - 104*n^2 + 108*n - 20)*x^3)*D[P[n - 2, x], x] - (n*x^2 - (8*n^2 - 8*n + 2)*x^3 + (16*n^2 - 20*n + 4)*x^4)*D[P[n - 2, x], {x, 2}] + (n - 1 + (8*n^2 - 16*n + 8)*x)*P[n - 1, x] + ((2*n - 1)*x - (8*n^2 - 8*n)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = (-((8*n^2 - 16*n)*x - (36*n^3 - 116*n^2 + 96*n - 16)*x^2)*v[i - 2] - (2*n*x - (8*n^2 + 2)*x^2 - (16*n^3 - 104*n^2 + 108*n - 20)*x^3)*deriv(v[i - 2], x) - (n*x^2 - (8*n^2 - 8*n + 2)*x^3 + (16*n^2 - 20*n + 4)*x^4)*deriv(deriv(v[i - 2], x), x) + (n - 1 + (8*n^2 - 16*n + 8)*x)*v[i - 1] + ((2*n - 1)*x - (8*n^2 - 8*n)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, 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 ≥ 2
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.