A253283 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(5 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-9\,x^{2} + 9\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(30 + 16\,x - 32\,n - 16\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||
| 2 | 1 | 1 | ||||||
| 3 | 3 | 2 | 5 | |||||
| 4 | 10 | 12 | 3 | 25 | ||||
| 5 | 35 | 60 | 30 | 4 | 129 | |||
| 6 | 126 | 280 | 210 | 60 | 5 | 681 | ||
| 7 | 462 | 1260 | 1260 | 560 | 105 | 6 | 3653 | |
| 8 | 1716 | 5544 | 6930 | 4200 | 1260 | 168 | 7 | 19825 |
Compute more
?
Technical details
- Node id
n-133f3a16dadf73ac· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-2c1fea4a558d4e6c
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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; 3: 3, 2; 4: 10, 12, 3; 5: 35, 60, 30, 4; 6: 126, 280, 210, 60, 5; 7: 462, 1260, 1260, 560, 105, 6; 8: 1716, 5544, 6930, 4200, 1260, 168, 7; 9: 6435, 24024, 36036, 27720, 11550, 2520, 252, 8; 10: 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;
%F FormulaThe row polynomials satisfy (2*n^2-7*n+5)*P(n,x) = -(2*n^2-9*n+9)*x^2*P(n-2,x) + (8*n^2-32*n+30+(4*n^2-16*n+16)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 9)*x^2*P[n - 2, x] + (8*n^2 - 32*n + 30 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x])/(2*n^2 - 7*n + 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 9*n + 9)*x^2*v[i - 2] + (8*n^2 - 32*n + 30 + (4*n^2 - 16*n + 16)*x)*v[i - 1])/(2*n^2 - 7*n + 5)); 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: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.?
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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: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.?
Related ?
Same fingerprint ?
- A001263 → θ+1 → x ↦ x+1
- A253283: Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.
- A367270 → x ↦ x+1
- A360282: Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n >…
- A123160 → D
- A360282 → reverse
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.