derivative of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - 3\,x + 2\,n x\right)\,P_{n}(x) = \left(-2 - 2\,x + 6\,x^{2} + 14\,x^{3} - 28\,x^{4} + 12\,x^{5} - 4\,n x - 10\,n x^{2} - 2\,n x^{3} + 54\,n x^{4} - 38\,n x^{5} - 4\,n^{2} x^{3} - 20\,n^{2} x^{4} + 32\,n^{2} x^{5} - 8\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-x - x^{2} + 6\,x^{3} - 2\,x^{4} - 5\,x^{5} + 3\,x^{6} - 2\,n x^{2} - 6\,n x^{3} + 10\,n x^{4} + 6\,n x^{5} - 8\,n x^{6} - 4\,n^{2} x^{4} + 4\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(3 - 3\,x - 15\,x^{2} + 15\,x^{3} + 6\,n x + 12\,n x^{2} - 22\,n x^{3} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 3\,x^{2} - x^{3} + 3\,x^{4} + 2\,n x^{2} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | ||||||||||||||
| 1 | 2 | 2 | 4 | ||||||||||||
| 2 | 6 | 20 | 18 | 4 | 48 | ||||||||||
| 3 | 14 | 94 | 204 | 188 | 70 | 6 | 576 | ||||||||
| 4 | 30 | 352 | 1350 | 2424 | 2250 | 1056 | 210 | 8 | 7680 | ||||||
| 5 | 62 | 1194 | 7176 | 20648 | 33060 | 30972 | 16744 | 4776 | 558 | 10 | 115200 | ||||
| 6 | 126 | 3852 | 34146 | 143868 | 345660 | 512952 | 483924 | 287736 | 102438 | 19260 | 1386 | 12 | 1935360 | ||
| 7 | 254 | 12086 | 152676 | 895140 | 3002730 | 6362466 | 8939672 | 8483288 | 5404914 | 2237850 | 559812 | 72516 | 3302 | 14 | 36126720 |
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Technical details
- Node id
n-95696ed532f4af38· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-435ee979f8f45497
Observed on the stored rows ?
- Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 2, 2; 2: 6, 20, 18, 4; 3: 14, 94, 204, 188, 70, 6; 4: 30, 352, 1350, 2424, 2250, 1056, 210, 8; 5: 62, 1194, 7176, 20648, 33060, 30972, 16744, 4776, 558, 10; 6: 126, 3852, 34146, 143868, 345660, 512952, 483924, 287736, 102438, 19260, 1386, 12; 7: 254, 12086, 152676, 895140, 3002730, 6362466, 8939672, 8483288, 5404914, 2237850, 559812, 72516, 3302, 14; 8: 510, 37304, 657630, 5197072, 23307750, 66513624, 128401350, 173132720, 165087450, 110856040, 51277050, 15591216, 2849730, 261128, 7650, 16; 9: 1022, 113986, 2766912, 28858160, 168183880, 622876872, 1573979456, 2827216432, 3691297620, 3534020540, 2473396288, 1245753744, 437278088, 101003560, 13834560, 911888, 17374, 18;
%F FormulaThe row polynomials satisfy (1+(2*n-3)*x)*P(n,x) = -(2+(4*n+2)*x+(10*n-6)*x^2+(4*n^2+2*n-14)*x^3+(20*n^2-54*n+28)*x^4+(8*n^3-32*n^2+38*n-12)*x^5)*P(n-2,x) - (x+(2*n+1)*x^2+(6*n-6)*x^3+(4*n^2-10*n+2)*x^4-(6*n-5)*x^5-(4*n^2-8*n+3)*x^6)*d/dx P(n-2,x) + (3+(6*n-3)*x+(12*n-15)*x^2+(8*n^2-22*n+15)*x^3)*P(n-1,x) + (x+(2*n-3)*x^2-x^3-(2*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 2*x.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 + (4*n + 2)*x + (10*n - 6)*x^2 + (4*n^2 + 2*n - 14)*x^3 + (20*n^2 - 54*n + 28)*x^4 + (8*n^3 - 32*n^2 + 38*n - 12)*x^5)*P[n - 2, x] - (x + (2*n + 1)*x^2 + (6*n - 6)*x^3 + (4*n^2 - 10*n + 2)*x^4 - (6*n - 5)*x^5 - (4*n^2 - 8*n + 3)*x^6)*D[P[n - 2, x], x] + (3 + (6*n - 3)*x + (12*n - 15)*x^2 + (8*n^2 - 22*n + 15)*x^3)*P[n - 1, x] + (x + (2*n - 3)*x^2 - x^3 - (2*n - 3)*x^4)*D[P[n - 1, x], x])/(1 + (2*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (4*n + 2)*x + (10*n - 6)*x^2 + (4*n^2 + 2*n - 14)*x^3 + (20*n^2 - 54*n + 28)*x^4 + (8*n^3 - 32*n^2 + 38*n - 12)*x^5)*v[i - 2] - (x + (2*n + 1)*x^2 + (6*n - 6)*x^3 + (4*n^2 - 10*n + 2)*x^4 - (6*n - 5)*x^5 - (4*n^2 - 8*n + 3)*x^6)*deriv(v[i - 2], x) + (3 + (6*n - 3)*x + (12*n - 15)*x^2 + (8*n^2 - 22*n + 15)*x^3)*v[i - 1] + (x + (2*n - 3)*x^2 - x^3 - (2*n - 3)*x^4)*deriv(v[i - 1], x))/(1 + (2*n - 3)*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
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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Product of real-rooted factors (elementary) polytool ? step ›
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
- ↑ Product of real-rooted factors (elementary) (above)
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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.