A185263: Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.
A185263 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A185263
\(\left(2 + n\right)\,P_{n}(x) = \left(1 + 2\,n\right)\,P_{n-1}(x) + \left(1 - n - n^{2} x + n^{3} x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 2 | 1 | 1 | 2 | ||
| 3 | 1 | 5 | 6 | ||
| 4 | 1 | 15 | 8 | 24 | |
| 5 | 1 | 35 | 84 | 120 | |
| 6 | 1 | 70 | 469 | 180 | 720 |
| 7 | 1 | 126 | 1869 | 3044 | 5040 |
All 15 rows b-files: A000142 (row sums), A185263 ?
Compute more
?
Technical details
- Node id
n-136f914693403ae6· revision 0 · JSON- Definition source
oeis_catalog:A185263- Fingerprint
f6-fe5aabeaa01cd96d
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?
Row sums ? A104150, A159333, A354074, A000142, A370360, A124355, A338112, A358185
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1; 2: 1, 1; 3: 1, 5; 4: 1, 15, 8; 5: 1, 35, 84; 6: 1, 70, 469, 180; 7: 1, 126, 1869, 3044; 8: 1, 210, 5985, 26060, 8064; 9: 1, 330, 16401, 152900, 193248;
%F FormulaThe row polynomials satisfy (n+2)*P(n,x) = (2*n+1)*P(n-1,x) - (n-1-(n^3-n^2)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*P[n - 1, x] - (n - 1 - (n^3 - n^2)*x)*P[n - 2, x])/(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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*v[i - 1] - (n - 1 - (n^3 - n^2)*x)*v[i - 2])/(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
A185263 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1; 2: 1, 1; 3: 1, 5; 4: 1, 15, 8; 5: 1, 35, 84; 6: 1, 70, 469, 180; 7: 1, 126, 1869, 3044; 8: 1, 210, 5985, 26060, 8064; 9: 1, 330, 16401, 152900, 193248;
%F FormulaThe row polynomials satisfy (n+2)*P(n,x) = (2*n+1)*P(n-1,x) - (n-1-(n^3-n^2)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*P[n - 1, x] - (n - 1 - (n^3 - n^2)*x)*P[n - 2, x])/(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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*v[i - 1] - (n - 1 - (n^3 - n^2)*x)*v[i - 2])/(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)
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
Proof chains ? JSON
Used in 6
- basis_map_A013610 of Triangle T(n,k) read by rows: coefficients (in compressed … Image under a basis map u^k -> P_k (A013610)
- basis_map_A013611 of Triangle T(n,k) read by rows: coefficients (in compressed … Image under a basis map u^k -> P_k (A013611)
- shift_x_plus_one of Triangle T(n,k) read by rows: coefficients (in compressed f… Image under a basis map u^k -> P_k (A007318)
- basis_map_A013609 of Triangle T(n,k) read by rows: coefficients (in compressed … Image under a basis map u^k -> P_k (A013609)
- basis_map_A008277 of Triangle T(n,k) read by rows: coefficients (in compressed … Image under a basis map u^k -> P_k (A008277)
- shift_x_plus_one of Triangle T(n,k) read by rows: coefficients (in compressed f… Image under a real-rootedness preserver (affine, shift 0)
Related ?
Same fingerprint ?
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