A359363 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-18\,x - 12\,x^{2} + 6\,x^{3} - 6\,n - 33\,n x + 8\,n x^{2} + 5\,n x^{3} - 5\,n^{2} + 13\,n^{2} x^{2} + n^{2} x^{3} + 5\,n^{3} + 12\,n^{3} x + 3\,n^{3} x^{2} + 5\,n^{4} + 3\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(18\,x + 21\,x^{2} - 3\,x^{4} - 6\,n + 6\,n x - 20\,n x^{2} - 12\,n x^{3} + 5\,n x^{4} - 5\,n^{2} - 47\,n^{2} x - 55\,n^{2} x^{2} + 6\,n^{2} x^{3} + 10\,n^{2} x^{4} + 5\,n^{3} - 31\,n^{3} x + 30\,n^{3} x^{3} + 5\,n^{4} + 8\,n^{4} x + 30\,n^{4} x^{2} + n^{5} + 10\,n^{5} x\right)\,P_{n-1}(x) + \left(-36\,x - 69\,x^{2} - 27\,x^{3} + 9\,x^{4} + 3\,x^{5} - 15\,n x + 37\,n x^{2} + 79\,n x^{3} + 12\,n x^{4} - 15\,n x^{5} + 27\,n^{2} x + 83\,n^{2} x^{2} + 17\,n^{2} x^{3} - 45\,n^{2} x^{4} + 21\,n^{3} x - 45\,n^{3} x^{3} + 3\,n^{4} x - 15\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(-12\,x^{2} - 24\,x^{3} - 6\,x^{4} + 12\,x^{5} + 6\,x^{6} - 4\,n x^{2} + 8\,n x^{3} + 30\,n x^{4} + 18\,n x^{5} + 10\,n^{2} x^{2} + 28\,n^{2} x^{3} + 18\,n^{2} x^{4} + 6\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 1 | 2 | |||||||||||||
| 2 | 1 | 3 | 1 | 5 | ||||||||||||
| 3 | 1 | 9 | 7 | 1 | 18 | |||||||||||
| 4 | 1 | 21 | 40 | 14 | 1 | 77 | ||||||||||
| 5 | 1 | 41 | 170 | 130 | 25 | 1 | 368 | |||||||||
| 6 | 1 | 71 | 560 | 875 | 350 | 41 | 1 | 1899 | ||||||||
| 7 | 1 | 113 | 1526 | 4410 | 3430 | 826 | 63 | 1 | 10370 | |||||||
| 8 | 1 | 169 | 3612 | 17640 | 24696 | 11172 | 1764 | 92 | 1 | 59147 | ||||||
| 9 | 1 | 241 | 7680 | 58968 | 137592 | 109368 | 31752 | 3480 | 129 | 1 | 349212 | |||||
| 10 | 1 | 331 | 15015 | 171270 | 623700 | 814968 | 407484 | 81180 | 6435 | 175 | 1 | 2120560 | ||||
| 11 | 1 | 441 | 27445 | 444675 | 2395800 | 4848228 | 3933468 | 1328580 | 190575 | 11275 | 231 | 1 | 13180720 | |||
| 12 | 1 | 573 | 47476 | 1053910 | 8045895 | 23981958 | 29899584 | 16195608 | 3893175 | 416845 | 18876 | 298 | 1 | 83554200 | ||
| 13 | 1 | 729 | 78442 | 2316314 | 24191167 | 101897367 | 186249492 | 153858276 | 58709079 | 10449439 | 858858 | 30394 | 377 | 1 | 538639936 | |
| 14 | 1 | 911 | 124670 | 4778137 | 66314248 | 381476095 | 982533552 | 1184303835 | 684264438 | 191668477 | 26052026 | 1681043 | 47320 | 469 | 1 | 3523245223 |
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Technical details
- Node id
n-b0113a98618fdee8· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-d997120effc7e338
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, 1; 2: 1, 3, 1; 3: 1, 9, 7, 1; 4: 1, 21, 40, 14, 1; 5: 1, 41, 170, 130, 25, 1; 6: 1, 71, 560, 875, 350, 41, 1; 7: 1, 113, 1526, 4410, 3430, 826, 63, 1; 8: 1, 169, 3612, 17640, 24696, 11172, 1764, 92, 1; 9: 1, 241, 7680, 58968, 137592, 109368, 31752, 3480, 129, 1;
%F FormulaThe row polynomials satisfy (n^5+5*n^4+5*n^3-5*n^2-6*n+(3*n^4+12*n^3-33*n-18)*x+(3*n^3+13*n^2+8*n-12)*x^2+(n^2+5*n+6)*x^3)*P(n,x) = (n^5+5*n^4+5*n^3-5*n^2-6*n+(10*n^5+8*n^4-31*n^3-47*n^2+6*n+18)*x+(30*n^4-55*n^2-20*n+21)*x^2+(30*n^3+6*n^2-12*n)*x^3+(10*n^2+5*n-3)*x^4)*P(n-1,x) + ((3*n^4+21*n^3+27*n^2-15*n-36)*x-(15*n^4-83*n^2-37*n+69)*x^2-(45*n^3-17*n^2-79*n+27)*x^3-(45*n^2-12*n-9)*x^4-(15*n-3)*x^5)*d/dx P(n-1,x) + ((6*n^3+10*n^2-4*n-12)*x^2+(6*n^3+28*n^2+8*n-24)*x^3+(18*n^2+30*n-6)*x^4+(18*n+12)*x^5+6*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 8*n^4 - 31*n^3 - 47*n^2 + 6*n + 18)*x + (30*n^4 - 55*n^2 - 20*n + 21)*x^2 + (30*n^3 + 6*n^2 - 12*n)*x^3 + (10*n^2 + 5*n - 3)*x^4)*P[n - 1, x] + ((3*n^4 + 21*n^3 + 27*n^2 - 15*n - 36)*x - (15*n^4 - 83*n^2 - 37*n + 69)*x^2 - (45*n^3 - 17*n^2 - 79*n + 27)*x^3 - (45*n^2 - 12*n - 9)*x^4 - (15*n - 3)*x^5)*D[P[n - 1, x], x] + ((6*n^3 + 10*n^2 - 4*n - 12)*x^2 + (6*n^3 + 28*n^2 + 8*n - 24)*x^3 + (18*n^2 + 30*n - 6)*x^4 + (18*n + 12)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (3*n^4 + 12*n^3 - 33*n - 18)*x + (3*n^3 + 13*n^2 + 8*n - 12)*x^2 + (n^2 + 5*n + 6)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 8*n^4 - 31*n^3 - 47*n^2 + 6*n + 18)*x + (30*n^4 - 55*n^2 - 20*n + 21)*x^2 + (30*n^3 + 6*n^2 - 12*n)*x^3 + (10*n^2 + 5*n - 3)*x^4)*v[i - 1] + ((3*n^4 + 21*n^3 + 27*n^2 - 15*n - 36)*x - (15*n^4 - 83*n^2 - 37*n + 69)*x^2 - (45*n^3 - 17*n^2 - 79*n + 27)*x^3 - (45*n^2 - 12*n - 9)*x^4 - (15*n - 3)*x^5)*deriv(v[i - 1], x) + ((6*n^3 + 10*n^2 - 4*n - 12)*x^2 + (6*n^3 + 28*n^2 + 8*n - 24)*x^3 + (18*n^2 + 30*n - 6)*x^4 + (18*n + 12)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (3*n^4 + 12*n^3 - 33*n - 18)*x + (3*n^3 + 13*n^2 + 8*n - 12)*x^2 + (n^2 + 5*n + 6)*x^3)); 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 differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 …?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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from
Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - 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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.