A154986 → D+1 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(3 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-12 - 24\,x - 12\,x^{2} + 40\,n + 80\,n x + 40\,n x^{2} - 63\,n^{2} - 126\,n^{2} x - 63\,n^{2} x^{2} + 59\,n^{3} + 118\,n^{3} x + 59\,n^{3} x^{2} - 32\,n^{4} - 64\,n^{4} x - 32\,n^{4} x^{2} + 9\,n^{5} + 18\,n^{5} x + 9\,n^{5} x^{2} - n^{6} - 2\,n^{6} x - n^{6} x^{2}\right)\,P_{n-4}(x) + \left(-12 - 20\,x - 8\,x^{2} + 30\,n + 50\,n x + 20\,n x^{2} - 32\,n^{2} - 52\,n^{2} x - 20\,n^{2} x^{2} + 18\,n^{3} + 28\,n^{3} x + 10\,n^{3} x^{2} - 4\,n^{4} - 6\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-3}(x) + \left(-6 - 8\,x - 3\,x^{2} + 2\,n x + 3\,n x^{2} + 8\,n^{2} + 8\,n^{2} x - n^{2} x^{2} - 8\,n^{3} - 8\,n^{3} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-2}(x) + \left(10 + 6\,x - 10\,n - 6\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
\(P_{2}(x) = 12 + 8\,x + x^{2}\)
\(P_{3}(x) = 60 + 64\,x + 17\,x^{2} + x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 1 | 4 | ||||||
| 2 | 12 | 8 | 1 | 21 | |||||
| 3 | 60 | 64 | 17 | 1 | 142 | ||||
| 4 | 360 | 536 | 232 | 32 | 1 | 1161 | |||
| 5 | 2520 | 4872 | 3048 | 704 | 55 | 1 | 11200 | ||
| 6 | 20160 | 48144 | 39984 | 13720 | 1884 | 88 | 1 | 123981 | |
| 7 | 181440 | 517632 | 541776 | 254000 | 52940 | 4536 | 133 | 1 | 1552458 |
Compute more
?
Technical details
- Node id
n-38eda9406f68fa74· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-2c634a7d61516747
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: 3, 1; 2: 12, 8, 1; 3: 60, 64, 17, 1; 4: 360, 536, 232, 32, 1; 5: 2520, 4872, 3048, 704, 55, 1; 6: 20160, 48144, 39984, 13720, 1884, 88, 1; 7: 181440, 517632, 541776, 254000, 52940, 4536, 133, 1; 8: 1814400, 6029568, 7649280, 4613184, 1350288, 180816, 9968, 192, 1; 9: 19958400, 75818880, 113448960, 84442752, 32570928, 6251088, 554672, 20256, 267, 1;
%F FormulaThe row polynomials satisfy (n^2-3*n+3)*P(n,x) = -(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12+(2*n^6-18*n^5+64*n^4-118*n^3+126*n^2-80*n+24)*x+(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2)*P(n-4,x) - (4*n^4-18*n^3+32*n^2-30*n+12+(6*n^4-28*n^3+52*n^2-50*n+20)*x+(2*n^4-10*n^3+20*n^2-20*n+8)*x^2)*P(n-3,x) + (2*n^4-8*n^3+8*n^2-6+(2*n^4-8*n^3+8*n^2+2*n-8)*x-(n^2-3*n+3)*x^2)*P(n-2,x) + (4*n^2-10*n+10+(2*n^2-6*n+6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 12 + 8*x + x^2, P(3,x) = 60 + 64*x + 17*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 12 + 8*x + x^2;
P[3, x_] := 60 + 64*x + 17*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12 + (2*n^6 - 18*n^5 + 64*n^4 - 118*n^3 + 126*n^2 - 80*n + 24)*x + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2)*P[n - 4, x] - (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12 + (6*n^4 - 28*n^3 + 52*n^2 - 50*n + 20)*x + (2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x^2)*P[n - 3, x] + (2*n^4 - 8*n^3 + 8*n^2 - 6 + (2*n^4 - 8*n^3 + 8*n^2 + 2*n - 8)*x - (n^2 - 3*n + 3)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 10 + (2*n^2 - 6*n + 6)*x)*P[n - 1, x])/(n^2 - 3*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 12 + 8*x + x^2; v[4] = 60 + 64*x + 17*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12 + (2*n^6 - 18*n^5 + 64*n^4 - 118*n^3 + 126*n^2 - 80*n + 24)*x + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2)*v[i - 4] - (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12 + (6*n^4 - 28*n^3 + 52*n^2 - 50*n + 20)*x + (2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x^2)*v[i - 3] + (2*n^4 - 8*n^3 + 8*n^2 - 6 + (2*n^4 - 8*n^3 + 8*n^2 + 2*n - 8)*x - (n^2 - 3*n + 3)*x^2)*v[i - 2] + (4*n^2 - 10*n + 10 + (2*n^2 - 6*n + 6)*x)*v[i - 1])/(n^2 - 3*n + 3)); 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 (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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
one_plus_derivative of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2),…?
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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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 T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
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from
one_plus_derivative of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2),…?
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.