A290596 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 - 10\,x + 42\,x^{2} + 6\,n x - 39\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x^{2} + 48\,x^{3} - 18\,n x^{3}\right)\,P_{n-1}'(x) + \left(9\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 2 | 3 | ||||||
| 3 | 1 | 10 | 10 | 21 | |||||
| 4 | 1 | 24 | 120 | 80 | 225 | ||||
| 5 | 1 | 44 | 528 | 1760 | 880 | 3213 | |||
| 6 | 1 | 70 | 1540 | 12320 | 30800 | 12320 | 57051 | ||
| 7 | 1 | 102 | 3570 | 52360 | 314160 | 628320 | 209440 | 1207953 | |
| 8 | 1 | 140 | 7140 | 166600 | 1832600 | 8796480 | 14660800 | 4188800 | 29652561 |
Compute more
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Technical details
- Node id
n-b3547558d8fdf454· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-94f770baffeadbcb
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, 2; 3: 1, 10, 10; 4: 1, 24, 120, 80; 5: 1, 44, 528, 1760, 880; 6: 1, 70, 1540, 12320, 30800, 12320; 7: 1, 102, 3570, 52360, 314160, 628320, 209440; 8: 1, 140, 7140, 166600, 1832600, 8796480, 14660800, 4188800; 9: 1, 184, 12880, 437920, 7663600, 67439680, 269758720, 385369600, 96342400; 10: 1, 234, 21528, 1004640, 25618320, 358656480, 2630147520, 9017648640, 11272060800, 2504902400;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(6*n-10)*x+(9*n^2-39*n+42)*x^2)*P(n-1,x) - (6*x^2+(18*n-48)*x^3)*d/dx P(n-1,x) + 9*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n - 10)*x + (9*n^2 - 39*n + 42)*x^2)*P[n - 1, x] - (6*x^2 + (18*n - 48)*x^3)*D[P[n - 1, x], x] + 9*x^4*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (6*n - 10)*x + (9*n^2 - 39*n + 42)*x^2)*v[i - 1] - (6*x^2 + (18*n - 48)*x^3)*deriv(v[i - 1], x) + 9*x^4*deriv(deriv(v[i - 1], x), x)); 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].?
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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. A generalization of unsigned Lah numbers, called L[3,1].?
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.