A358623: Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.
A358623 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A358623
\(P_{n}(x) = \left(x\right)\,P_{n-1}'(x) + \left(-2\,x + n x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 0\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | |||
| 2 | 0 | ||||
| 3 | 0 | 1 | 1 | ||
| 4 | 0 | 1 | 1 | ||
| 5 | 0 | 1 | 3 | 4 | |
| 6 | 0 | 1 | 10 | 11 | |
| 7 | 0 | 1 | 25 | 15 | 41 |
| 8 | 0 | 1 | 56 | 105 | 162 |
All 15 rows b-files: A000296 (row sums), A008299 ?
Compute more
?
Technical details
- Node id
n-854413c64ac5ddd8· revision 0 · JSON- Definition source
oeis_catalog:A358623- Fingerprint
f6-e8edac49e7142220
Observed on the stored rows ?
- Degrees: 0, -1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, 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 ?
Row sums ? A000296
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 0; 3: 0, 1; 4: 0, 1; 5: 0, 1, 3; 6: 0, 1, 10; 7: 0, 1, 25, 15; 8: 0, 1, 56, 105; 9: 0, 1, 119, 490, 105; 10: 0, 1, 246, 1918, 1260;
%F FormulaThe row polynomials satisfy P(n,x) = x*d/dx P(n-1,x) + (n-2)*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 0.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 0;
P[n_, x_] := P[n, x] = Expand[x*D[P[n - 1, x], x] + (n - 2)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; for(n = 3, N, my(i = n); v[i] = x*deriv(v[i - 1], x) + (n - 2)*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
A008299 triangle, offset 2, from P3, low zeros dropped
%e ExampleTriangle begins: 2: 1; 3: 1; 4: 1, 3; 5: 1, 10; 6: 1, 25, 15; 7: 1, 56, 105; 8: 1, 119, 490, 105; 9: 1, 246, 1918, 1260; 10: 1, 501, 6825, 9450, 945; 11: 1, 1012, 22935, 56980, 17325;
%F FormulaRow n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = x*d/dx P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 4, with P(2,x) = x, P(3,x) = x.
%t MathematicaP[2, x_] := x;
P[3, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*D[P[n - 1, x], x] + (n - 1)*x*P[n - 2, x]];
Table[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 2, 11}] // Flatten%o PARI/GPP(N) = {my(v = vector(N - 1)); v[1] = x; v[2] = x; for(n = 4, N, my(i = n - 1); v[i] = x*deriv(v[i - 1], x) + (n - 1)*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};
T(11)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 3, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 3: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 3, 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 ≥ 4
Proof chains ? JSON
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