A103633 → basis A007318 → reverse
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 - 2\,x - 4\,x^{2} + 3\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
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 | 2 | 6 | ||||||||||||
| 3 | 1 | 4 | 5 | 2 | 12 | |||||||||||
| 4 | 1 | 6 | 13 | 12 | 4 | 36 | ||||||||||
| 5 | 1 | 7 | 19 | 25 | 16 | 4 | 72 | |||||||||
| 6 | 1 | 9 | 33 | 63 | 66 | 36 | 8 | 216 | ||||||||
| 7 | 1 | 10 | 42 | 96 | 129 | 102 | 44 | 8 | 432 | |||||||
| 8 | 1 | 12 | 62 | 180 | 321 | 360 | 248 | 96 | 16 | 1296 | ||||||
| 9 | 1 | 13 | 74 | 242 | 501 | 681 | 608 | 344 | 112 | 16 | 2592 | |||||
| 10 | 1 | 15 | 100 | 390 | 985 | 1683 | 1970 | 1560 | 800 | 240 | 32 | 7776 | ||||
| 11 | 1 | 16 | 115 | 490 | 1375 | 2668 | 3653 | 3530 | 2360 | 1040 | 272 | 32 | 15552 | |||
| 12 | 1 | 18 | 147 | 720 | 2355 | 5418 | 8989 | 10836 | 9420 | 5760 | 2352 | 576 | 64 | 46656 | ||
| 13 | 1 | 19 | 165 | 867 | 3075 | 7773 | 14407 | 19825 | 20256 | 15180 | 8112 | 2928 | 640 | 64 | 93312 | |
| 14 | 1 | 21 | 203 | 1197 | 4809 | 13923 | 29953 | 48639 | 59906 | 55692 | 38472 | 19152 | 6496 | 1344 | 128 | 279936 |
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Technical details
- Node id
n-45465ea314fc3ade· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-cd565eb4d46acae6
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): yes
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, 2; 3: 1, 4, 5, 2; 4: 1, 6, 13, 12, 4; 5: 1, 7, 19, 25, 16, 4; 6: 1, 9, 33, 63, 66, 36, 8; 7: 1, 10, 42, 96, 129, 102, 44, 8; 8: 1, 12, 62, 180, 321, 360, 248, 96, 16; 9: 1, 13, 74, 242, 501, 681, 608, 344, 112, 16;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(3*n-2)*x+(4*n-4)*x^2)*P(n-1,x) - (2*x+6*x^2+4*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 2)*x + (4*n - 4)*x^2)*P[n - 1, x] - (2*x + 6*x^2 + 4*x^3)*D[P[n - 1, x], x]];
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] = (1 + (3*n - 2)*x + (4*n - 4)*x^2)*v[i - 1] - (2*x + 6*x^2 + 4*x^3)*deriv(v[i - 1], x)); 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
Related ?
Same fingerprint ?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [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.