A249250 → basis A013609 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(3 + 3\,x - 11\,x^{2} - 11\,x^{3} - 2\,x^{4} - 3\,n - 15\,n x - 10\,n x^{2} + 7\,n x^{3} + 2\,n x^{4} + 6\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{2} x^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(3 + 3\,x - 11\,x^{2} - 11\,x^{3} - 2\,x^{4} + 15\,n + 21\,n x - 43\,n x^{2} - 53\,n x^{3} - 10\,n x^{4} - 9\,n^{2} - 39\,n^{2} x - 54\,n^{2} x^{2} - 10\,n^{2} x^{3} - 9\,n^{3} - 15\,n^{3} x\right)\,P_{n-1}(x) + \left(18 + 123\,x + 321\,x^{2} + 395\,x^{3} + 227\,x^{4} + 52\,x^{5} + 4\,x^{6} + 27\,n + 144\,n x + 267\,n x^{2} + 198\,n x^{3} + 50\,n x^{4} + 4\,n x^{5} + 9\,n^{2} + 36\,n^{2} x + 39\,n^{2} x^{2} + 6\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 10 | 6 | 16 | ||||||
| 2 | 34 | 48 | 12 | 94 | |||||
| 3 | 124 | 288 | 168 | 24 | 604 | ||||
| 4 | 412 | 1344 | 1368 | 480 | 48 | 3652 | |||
| 5 | 1336 | 5640 | 8304 | 5088 | 1248 | 96 | 21712 | ||
| 6 | 4216 | 21888 | 42528 | 38400 | 16416 | 3072 | 192 | 126712 | |
| 7 | 13072 | 80640 | 195264 | 235584 | 149184 | 48384 | 7296 | 384 | 729808 |
Compute more
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Technical details
- Node id
n-ea06c2ded5f0690b· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-4635ca94cfd32645
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: 10, 6; 2: 34, 48, 12; 3: 124, 288, 168, 24; 4: 412, 1344, 1368, 480, 48; 5: 1336, 5640, 8304, 5088, 1248, 96; 6: 4216, 21888, 42528, 38400, 16416, 3072, 192; 7: 13072, 80640, 195264, 235584, 149184, 48384, 7296, 384; 8: 39952, 285696, 830016, 1260288, 1074240, 516096, 134016, 16896, 768; 9: 120736, 982368, 3332736, 6117120, 6604416, 4284288, 1644288, 354816, 38400, 1536;
%F FormulaThe row polynomials satisfy -(3*n-3-(3*n^3+6*n^2-15*n+3)*x-(12*n^2-10*n-11)*x^2-(2*n^2+7*n-11)*x^3-(2*n-2)*x^4)*P(n,x) = -(9*n^3+9*n^2-15*n-3+(15*n^3+39*n^2-21*n-3)*x+(54*n^2+43*n+11)*x^2+(10*n^2+53*n+11)*x^3+(10*n+2)*x^4)*P(n-1,x) + (9*n^2+27*n+18+(36*n^2+144*n+123)*x+(39*n^2+267*n+321)*x^2+(6*n^2+198*n+395)*x^3+(50*n+227)*x^4+(4*n+52)*x^5+4*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^3 + 9*n^2 - 15*n - 3 + (15*n^3 + 39*n^2 - 21*n - 3)*x + (54*n^2 + 43*n + 11)*x^2 + (10*n^2 + 53*n + 11)*x^3 + (10*n + 2)*x^4)*P[n - 1, x] + (9*n^2 + 27*n + 18 + (36*n^2 + 144*n + 123)*x + (39*n^2 + 267*n + 321)*x^2 + (6*n^2 + 198*n + 395)*x^3 + (50*n + 227)*x^4 + (4*n + 52)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-3*n + 3 + (3*n^3 + 6*n^2 - 15*n + 3)*x + (12*n^2 - 10*n - 11)*x^2 + (2*n^2 + 7*n - 11)*x^3 + (2*n - 2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(9*n^3 + 9*n^2 - 15*n - 3 + (15*n^3 + 39*n^2 - 21*n - 3)*x + (54*n^2 + 43*n + 11)*x^2 + (10*n^2 + 53*n + 11)*x^3 + (10*n + 2)*x^4)*v[i - 1] + (9*n^2 + 27*n + 18 + (36*n^2 + 144*n + 123)*x + (39*n^2 + 267*n + 321)*x^2 + (6*n^2 + 198*n + 395)*x^3 + (50*n + 227)*x^4 + (4*n + 52)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(-3*n + 3 + (3*n^3 + 6*n^2 - 15*n + 3)*x + (12*n^2 - 10*n - 11)*x^2 + (2*n^2 + 7*n - 11)*x^3 + (2*n - 2)*x^4)); 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
basis_map_A013609 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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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 ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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from
basis_map_A013609 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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.