A008292 → γ (shift +1) → x ↦ x+1 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(3 + 8\,x + 2\,n\right)\,P_{n}(x) = \left(-45 - 147\,x - 80\,x^{2} + 144\,x^{3} + 128\,x^{4} + 52\,n + 202\,n x + 218\,n x^{2} + 8\,n x^{3} - 64\,n x^{4} + 8\,n^{2} - 24\,n^{2} x - 80\,n^{2} x^{2} - 48\,n^{2} x^{3} - 8\,n^{3} - 16\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(45\,x + 237\,x^{2} + 464\,x^{3} + 400\,x^{4} + 128\,x^{5} + 30\,n + 166\,n x + 336\,n x^{2} + 296\,n x^{3} + 96\,n x^{4} + 12\,n^{2} + 40\,n^{2} x + 44\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-24 - 79\,x - 36\,x^{2} + 32\,x^{3} + 34\,n x + 40\,n x^{2} + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 45\,x - 68\,x^{2} - 32\,x^{3} - 6\,n - 14\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 2 | 5 | 2 | 7 | ||
| 3 | 17 | 8 | 25 | ||
| 4 | 93 | 86 | 16 | 195 | |
| 5 | 513 | 596 | 136 | 1245 | |
| 6 | 3477 | 5442 | 2352 | 272 | 11543 |
| 7 | 25569 | 48240 | 26880 | 3968 | 104657 |
Compute more
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Technical details
- Node id
n-582f3fb7e2f120e7· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-6e5ef435951ceba9
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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; 2: 5, 2; 3: 17, 8; 4: 93, 86, 16; 5: 513, 596, 136; 6: 3477, 5442, 2352, 272; 7: 25569, 48240, 26880, 3968; 8: 212733, 482670, 360288, 97792, 7936; 9: 1929393, 5065740, 4605768, 1645312, 176896;
%F FormulaThe row polynomials satisfy (2*n+3+8*x)*P(n,x) = -(8*n^3-8*n^2-52*n+45+(16*n^3+24*n^2-202*n+147)*x+(8*n^3+80*n^2-218*n+80)*x^2+(48*n^2-8*n-144)*x^3+(64*n-128)*x^4)*P(n-2,x) + (12*n^2+30*n+(40*n^2+166*n+45)*x+(44*n^2+336*n+237)*x^2+(16*n^2+296*n+464)*x^3+(96*n+400)*x^4+128*x^5)*d/dx P(n-2,x) + (8*n^2-24+(8*n^2+34*n-79)*x+(40*n-36)*x^2+32*x^3)*P(n-1,x) - (6*n+9+(14*n+45)*x+(8*n+68)*x^2+32*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 8*n^2 - 52*n + 45 + (16*n^3 + 24*n^2 - 202*n + 147)*x + (8*n^3 + 80*n^2 - 218*n + 80)*x^2 + (48*n^2 - 8*n - 144)*x^3 + (64*n - 128)*x^4)*P[n - 2, x] + (12*n^2 + 30*n + (40*n^2 + 166*n + 45)*x + (44*n^2 + 336*n + 237)*x^2 + (16*n^2 + 296*n + 464)*x^3 + (96*n + 400)*x^4 + 128*x^5)*D[P[n - 2, x], x] + (8*n^2 - 24 + (8*n^2 + 34*n - 79)*x + (40*n - 36)*x^2 + 32*x^3)*P[n - 1, x] - (6*n + 9 + (14*n + 45)*x + (8*n + 68)*x^2 + 32*x^3)*D[P[n - 1, x], x])/(2*n + 3 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 8*n^2 - 52*n + 45 + (16*n^3 + 24*n^2 - 202*n + 147)*x + (8*n^3 + 80*n^2 - 218*n + 80)*x^2 + (48*n^2 - 8*n - 144)*x^3 + (64*n - 128)*x^4)*v[i - 2] + (12*n^2 + 30*n + (40*n^2 + 166*n + 45)*x + (44*n^2 + 336*n + 237)*x^2 + (16*n^2 + 296*n + 464)*x^3 + (96*n + 400)*x^4 + 128*x^5)*deriv(v[i - 2], x) + (8*n^2 - 24 + (8*n^2 + 34*n - 79)*x + (40*n - 36)*x^2 + 32*x^3)*v[i - 1] - (6*n + 9 + (14*n + 45)*x + (8*n + 68)*x^2 + 32*x^3)*deriv(v[i - 1], x))/(2*n + 3 + 8*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]
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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
shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).?
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from
shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n…?
References
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.