A136157 → basis A007318
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 - x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)
\(P_{1}(x) = 4 + 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 3 | 3 | |||||||
| 1 | 4 | 3 | 7 | ||||||
| 2 | 4 | 7 | 3 | 14 | |||||
| 3 | 4 | 11 | 10 | 3 | 28 | ||||
| 4 | 4 | 15 | 21 | 13 | 3 | 56 | |||
| 5 | 4 | 19 | 36 | 34 | 16 | 3 | 112 | ||
| 6 | 4 | 23 | 55 | 70 | 50 | 19 | 3 | 224 | |
| 7 | 4 | 27 | 78 | 125 | 120 | 69 | 22 | 3 | 448 |
Compute more
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Technical details
- Node id
n-14b6be09569eb319· revision 0 · JSON- Definition source
derived:basis_map_A007318- Fingerprint
f6-59885d546c14be4d
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: 3; 1: 4, 3; 2: 4, 7, 3; 3: 4, 11, 10, 3; 4: 4, 15, 21, 13, 3; 5: 4, 19, 36, 34, 16, 3; 6: 4, 23, 55, 70, 50, 19, 3; 7: 4, 27, 78, 125, 120, 69, 22, 3; 8: 4, 31, 105, 203, 245, 189, 91, 25, 3; 9: 4, 35, 136, 308, 448, 434, 280, 116, 28, 3;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 4 + 3*x.
%t MathematicaP[0, x_] := 3;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(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]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.?
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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 by columns, (3, 1, 0, 0, 0, ...) in every column.?
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.