A078812 → reverse → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2 + 20\,x - 8\,x^{2} + 10\,n - 108\,n x + 56\,n x^{2} - 16\,n^{2} + 208\,n^{2} x - 152\,n^{2} x^{2} + 8\,n^{3} - 168\,n^{3} x + 200\,n^{3} x^{2} + 48\,n^{4} x - 128\,n^{4} x^{2} + 32\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-2 - 12\,x - 24\,x^{2} - 32\,x^{3} + 10\,n + 44\,n x + 24\,n x^{2} - 16\,n x^{3} - 16\,n^{2} - 16\,n^{2} x + 276\,n^{2} x^{2} + 176\,n^{2} x^{3} + 8\,n^{3} - 72\,n^{3} x - 424\,n^{3} x^{2} - 72\,n^{3} x^{3} + 48\,n^{4} x + 112\,n^{4} x^{2} - 160\,n^{4} x^{3} + 32\,n^{5} x^{2} + 96\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 12\,x^{3} - 16\,x^{4} + 4\,n x + 16\,n x^{2} - 4\,n x^{3} - 16\,n x^{4} - 4\,n^{2} x + 8\,n^{2} x^{2} + 116\,n^{2} x^{3} + 80\,n^{2} x^{4} - 24\,n^{3} x^{2} - 96\,n^{3} x^{3} - 16\,n^{4} x^{3} - 64\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 8 | 9 | ||||||
| 3 | 1 | 16 | 27 | 44 | |||||
| 4 | 1 | 24 | 90 | 64 | 179 | ||||
| 5 | 1 | 32 | 189 | 320 | 125 | 667 | |||
| 6 | 1 | 40 | 324 | 896 | 875 | 216 | 2352 | ||
| 7 | 1 | 48 | 495 | 1920 | 3150 | 2016 | 343 | 7973 | |
| 8 | 1 | 56 | 702 | 3520 | 8250 | 9072 | 4116 | 512 | 26229 |
Compute more
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Technical details
- Node id
n-84e3e93d54635aef· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-ae96443b231c3935
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, 8; 3: 1, 16, 27; 4: 1, 24, 90, 64; 5: 1, 32, 189, 320, 125; 6: 1, 40, 324, 896, 875, 216; 7: 1, 48, 495, 1920, 3150, 2016, 343; 8: 1, 56, 702, 3520, 8250, 9072, 4116, 512; 9: 1, 64, 945, 5824, 17875, 28512, 22638, 7680, 729; 10: 1, 72, 1224, 8960, 34125, 72072, 84084, 50688, 13365, 1000;
%F FormulaThe row polynomials satisfy (8*n^3-16*n^2+10*n-2+(48*n^4-168*n^3+208*n^2-108*n+20)*x+(32*n^5-128*n^4+200*n^3-152*n^2+56*n-8)*x^2)*P(n,x) = (8*n^3-16*n^2+10*n-2+(48*n^4-72*n^3-16*n^2+44*n-12)*x+(32*n^5+112*n^4-424*n^3+276*n^2+24*n-24)*x^2+(96*n^5-160*n^4-72*n^3+176*n^2-16*n-32)*x^3)*P(n-1,x) - ((4*n^2-4*n+1)*x+(24*n^3-8*n^2-16*n+6)*x^2+(16*n^4+96*n^3-116*n^2+4*n+12)*x^3+(64*n^4-80*n^2+16*n+16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 72*n^3 - 16*n^2 + 44*n - 12)*x + (32*n^5 + 112*n^4 - 424*n^3 + 276*n^2 + 24*n - 24)*x^2 + (96*n^5 - 160*n^4 - 72*n^3 + 176*n^2 - 16*n - 32)*x^3)*P[n - 1, x] - ((4*n^2 - 4*n + 1)*x + (24*n^3 - 8*n^2 - 16*n + 6)*x^2 + (16*n^4 + 96*n^3 - 116*n^2 + 4*n + 12)*x^3 + (64*n^4 - 80*n^2 + 16*n + 16)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 168*n^3 + 208*n^2 - 108*n + 20)*x + (32*n^5 - 128*n^4 + 200*n^3 - 152*n^2 + 56*n - 8)*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] = ((8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 72*n^3 - 16*n^2 + 44*n - 12)*x + (32*n^5 + 112*n^4 - 424*n^3 + 276*n^2 + 24*n - 24)*x^2 + (96*n^5 - 160*n^4 - 72*n^3 + 176*n^2 - 16*n - 32)*x^3)*v[i - 1] - ((4*n^2 - 4*n + 1)*x + (24*n^3 - 8*n^2 - 16*n + 6)*x^2 + (16*n^4 + 96*n^3 - 116*n^2 + 4*n + 12)*x^3 + (64*n^4 - 80*n^2 + 16*n + 16)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 168*n^3 + 208*n^2 - 108*n + 20)*x + (32*n^5 - 128*n^4 + 200*n^3 - 152*n^2 + 56*n - 8)*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
reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
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