A163936 → reverse → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2 - 4\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-12\,x + 12\,x^{2} + 8\,n x - 140\,n x^{2} + 316\,n^{2} x^{2} - 260\,n^{3} x^{2} + 72\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(x - 7\,x^{2} + 12\,x^{3} - 6\,n x^{2} - 128\,n x^{3} + 8\,n^{2} x^{2} + 252\,n^{2} x^{3} - 168\,n^{3} x^{3} + 32\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{2} - 3\,x^{3} + 2\,x^{4} + 2\,n x^{2} + 18\,n x^{3} - 20\,n x^{4} - 2\,n^{2} x^{2} - 32\,n^{2} x^{3} + 34\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-8\,x + 2\,n + 32\,n x - 2\,n^{2} - 40\,n^{2} x + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 6\,n x - 16\,n x^{2} + 4\,n^{2} x + 32\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 0 | 4 | 4 | ||||||
| 2 | 0 | 4 | 18 | 22 | |||||
| 3 | 0 | 4 | 72 | 96 | 172 | ||||
| 4 | 0 | 4 | 198 | 928 | 600 | 1730 | |||
| 5 | 0 | 4 | 468 | 5248 | 11100 | 4320 | 21140 | ||
| 6 | 0 | 4 | 1026 | 23232 | 110000 | 133488 | 35280 | 303030 | |
| 7 | 0 | 4 | 2160 | 89760 | 803000 | 2093040 | 1665216 | 322560 | 4975740 |
Compute more
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Technical details
- Node id
n-2116da0205a6248b· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-1957e2ef5036f48f
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 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: 0, 4; 2: 0, 4, 18; 3: 0, 4, 72, 96; 4: 0, 4, 198, 928, 600; 5: 0, 4, 468, 5248, 11100, 4320; 6: 0, 4, 1026, 23232, 110000, 133488, 35280; 7: 0, 4, 2160, 89760, 803000, 2093040, 1665216, 322560; 8: 0, 4, 4446, 319200, 4895000, 23184720, 38479896, 21832704, 3265920; 9: 0, 4, 9036, 1076160, 26562500, 207558000, 609563136, 705682944, 302447520, 36288000;
%F FormulaThe row polynomials satisfy (2*n^2-4*n+2)*P(n,x) = ((8*n-12)*x+(72*n^4-260*n^3+316*n^2-140*n+12)*x^2)*P(n-2,x) + (x+(8*n^2-6*n-7)*x^2+(32*n^4-168*n^3+252*n^2-128*n+12)*x^3)*d/dx P(n-2,x) - ((2*n^2-2*n-1)*x^2-(16*n^3-32*n^2+18*n-3)*x^3+(16*n^3-34*n^2+20*n-2)*x^4)*d^2/dx^2 P(n-2,x) - (2*n^2-2*n-(16*n^3-40*n^2+32*n-8)*x)*P(n-1,x) + ((4*n^2-6*n+1)*x-(16*n^3-32*n^2+16*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(((8*n - 12)*x + (72*n^4 - 260*n^3 + 316*n^2 - 140*n + 12)*x^2)*P[n - 2, x] + (x + (8*n^2 - 6*n - 7)*x^2 + (32*n^4 - 168*n^3 + 252*n^2 - 128*n + 12)*x^3)*D[P[n - 2, x], x] - ((2*n^2 - 2*n - 1)*x^2 - (16*n^3 - 32*n^2 + 18*n - 3)*x^3 + (16*n^3 - 34*n^2 + 20*n - 2)*x^4)*D[P[n - 2, x], {x, 2}] - (2*n^2 - 2*n - (16*n^3 - 40*n^2 + 32*n - 8)*x)*P[n - 1, x] + ((4*n^2 - 6*n + 1)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 4*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((8*n - 12)*x + (72*n^4 - 260*n^3 + 316*n^2 - 140*n + 12)*x^2)*v[i - 2] + (x + (8*n^2 - 6*n - 7)*x^2 + (32*n^4 - 168*n^3 + 252*n^2 - 128*n + 12)*x^3)*deriv(v[i - 2], x) - ((2*n^2 - 2*n - 1)*x^2 - (16*n^3 - 32*n^2 + 18*n - 3)*x^3 + (16*n^3 - 34*n^2 + 20*n - 2)*x^4)*deriv(deriv(v[i - 2], x), x) - (2*n^2 - 2*n - (16*n^3 - 40*n^2 + 32*n - 8)*x)*v[i - 1] + ((4*n^2 - 6*n + 1)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 4*n + 2)); 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 multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, 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
E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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.