A086645 → D → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(10 - 590\,x + 1000\,x^{2} - 9\,n + 1021\,n x - 2700\,n x^{2} + 2\,n^{2} - 659\,n^{2} x + 3030\,n^{2} x^{2} + 188\,n^{3} x - 1809\,n^{3} x^{2} - 20\,n^{4} x + 606\,n^{4} x^{2} - 108\,n^{5} x^{2} + 8\,n^{6} x^{2}\right)\,P_{n}(x) = \left(3 - 109\,x - 541\,x^{2} + 227\,x^{3} - 5\,n + 350\,n x + 1200\,n x^{2} - 1117\,n x^{3} + 2\,n^{2} - 369\,n^{2} x - 632\,n^{2} x^{2} + 2115\,n^{2} x^{3} + 148\,n^{3} x - 269\,n^{3} x^{2} - 1987\,n^{3} x^{3} - 20\,n^{4} x + 326\,n^{4} x^{2} + 982\,n^{4} x^{3} - 92\,n^{5} x^{2} - 244\,n^{5} x^{3} + 8\,n^{6} x^{2} + 24\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x + 102\,x^{2} - 132\,x^{3} + 34\,x^{4} + 4\,n x - 238\,n x^{2} + 420\,n x^{3} - 186\,n x^{4} + 176\,n^{2} x^{2} - 524\,n^{2} x^{3} + 348\,n^{2} x^{4} - 40\,n^{3} x^{2} + 332\,n^{3} x^{3} - 292\,n^{3} x^{4} - 112\,n^{4} x^{3} + 112\,n^{4} x^{4} + 16\,n^{5} x^{3} - 16\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0 | |||||||
| 2 | 1 | 1 | ||||||
| 3 | 6 | 8 | 14 | |||||
| 4 | 15 | 120 | 27 | 162 | ||||
| 5 | 28 | 560 | 756 | 64 | 1408 | |||
| 6 | 45 | 1680 | 5670 | 2880 | 125 | 10400 | ||
| 7 | 66 | 3960 | 24948 | 31680 | 8250 | 216 | 69120 | |
| 8 | 91 | 8008 | 81081 | 192192 | 125125 | 19656 | 343 | 426496 |
Compute more
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Technical details
- Node id
n-fbd62aba6d47a176· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-2a9ba3be646fcc5b
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
- 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: 0; 2: 1; 3: 6, 8; 4: 15, 120, 27; 5: 28, 560, 756, 64; 6: 45, 1680, 5670, 2880, 125; 7: 66, 3960, 24948, 31680, 8250, 216; 8: 91, 8008, 81081, 192192, 125125, 19656, 343; 9: 120, 14560, 216216, 823680, 1001000, 393120, 41160, 512; 10: 153, 24480, 501228, 2800512, 5469750, 4009824, 1049580, 78336, 729;
%F FormulaThe row polynomials satisfy (2*n^2-9*n+10-(20*n^4-188*n^3+659*n^2-1021*n+590)*x+(8*n^6-108*n^5+606*n^4-1809*n^3+3030*n^2-2700*n+1000)*x^2)*P(n,x) = (2*n^2-5*n+3-(20*n^4-148*n^3+369*n^2-350*n+109)*x+(8*n^6-92*n^5+326*n^4-269*n^3-632*n^2+1200*n-541)*x^2+(24*n^6-244*n^5+982*n^4-1987*n^3+2115*n^2-1117*n+227)*x^3)*P(n-1,x) + ((4*n-4)*x-(40*n^3-176*n^2+238*n-102)*x^2+(16*n^5-112*n^4+332*n^3-524*n^2+420*n-132)*x^3-(16*n^5-112*n^4+292*n^3-348*n^2+186*n-34)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t MathematicaP[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 - (20*n^4 - 148*n^3 + 369*n^2 - 350*n + 109)*x + (8*n^6 - 92*n^5 + 326*n^4 - 269*n^3 - 632*n^2 + 1200*n - 541)*x^2 + (24*n^6 - 244*n^5 + 982*n^4 - 1987*n^3 + 2115*n^2 - 1117*n + 227)*x^3)*P[n - 1, x] + ((4*n - 4)*x - (40*n^3 - 176*n^2 + 238*n - 102)*x^2 + (16*n^5 - 112*n^4 + 332*n^3 - 524*n^2 + 420*n - 132)*x^3 - (16*n^5 - 112*n^4 + 292*n^3 - 348*n^2 + 186*n - 34)*x^4)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10 - (20*n^4 - 188*n^3 + 659*n^2 - 1021*n + 590)*x + (8*n^6 - 108*n^5 + 606*n^4 - 1809*n^3 + 3030*n^2 - 2700*n + 1000)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - 5*n + 3 - (20*n^4 - 148*n^3 + 369*n^2 - 350*n + 109)*x + (8*n^6 - 92*n^5 + 326*n^4 - 269*n^3 - 632*n^2 + 1200*n - 541)*x^2 + (24*n^6 - 244*n^5 + 982*n^4 - 1987*n^3 + 2115*n^2 - 1117*n + 227)*x^3)*v[i - 1] + ((4*n - 4)*x - (40*n^3 - 176*n^2 + 238*n - 102)*x^2 + (16*n^5 - 112*n^4 + 332*n^3 - 524*n^2 + 420*n - 132)*x^3 - (16*n^5 - 112*n^4 + 292*n^3 - 348*n^2 + 186*n - 34)*x^4)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10 - (20*n^4 - 188*n^3 + 659*n^2 - 1021*n + 590)*x + (8*n^6 - 108*n^5 + 606*n^4 - 1809*n^3 + 3030*n^2 - 2700*n + 1000)*x^2)); 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]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
derivative of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle read by rows: T(n, k) = binomial(2n, 2k).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Triangle read by rows: T(n, k) = binomial(2n, 2k).?
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from
derivative of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
Related ?
Same fingerprint ?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023