A080248: Stirling-like number triangle defined by sequence A000217.
A080248 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A080248
\(\left(2\right)\,P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 1 | 2 | ||||||
| 2 | 1 | 4 | 1 | 6 | |||||
| 3 | 1 | 13 | 10 | 1 | 25 | ||||
| 4 | 1 | 40 | 73 | 20 | 1 | 135 | |||
| 5 | 1 | 121 | 478 | 273 | 35 | 1 | 909 | ||
| 6 | 1 | 364 | 2989 | 3208 | 798 | 56 | 1 | 7417 | |
| 7 | 1 | 1093 | 18298 | 35069 | 15178 | 1974 | 84 | 1 | 71698 |
All 15 rows b-files: A080248 ?
Compute more
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Technical details
- Node id
n-15df20466f2d1f05· revision 0 · JSON- Definition source
oeis_catalog:A080248- Fingerprint
f6-8583efd7a1bc8a2a
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 ?
Row sums ? A124373
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 4, 1; 3: 1, 13, 10, 1; 4: 1, 40, 73, 20, 1; 5: 1, 121, 478, 273, 35, 1; 6: 1, 364, 2989, 3208, 798, 56, 1; 7: 1, 1093, 18298, 35069, 15178, 1974, 84, 1; 8: 1, 3280, 110881, 368988, 262739, 56632, 4326, 120, 1; 9: 1, 9841, 668566, 3800761, 4310073, 1452011, 177760, 8646, 165, 1;
%F FormulaThe row polynomials satisfy 2*P(n,x) = (2+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) + x^2*d^2/dx^2 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[((2 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}])/(2)]];
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] = ((2 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x))/(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
A080248 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 4, 1; 3: 1, 13, 10, 1; 4: 1, 40, 73, 20, 1; 5: 1, 121, 478, 273, 35, 1; 6: 1, 364, 2989, 3208, 798, 56, 1; 7: 1, 1093, 18298, 35069, 15178, 1974, 84, 1; 8: 1, 3280, 110881, 368988, 262739, 56632, 4326, 120, 1; 9: 1, 9841, 668566, 3800761, 4310073, 1452011, 177760, 8646, 165, 1;
%F FormulaThe row polynomials satisfy 2*P(n,x) = (2+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) + x^2*d^2/dx^2 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[((2 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}])/(2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = ((2 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x))/(2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 Iterated real-rootedness preservers
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers
Proof chains ? JSON
Used in 7
- derivative of Stirling-like number triangle defined by sequence A000217. Image under a real-rootedness preserver (derivative polynomial, shift 0)
- reversal of Stirling-like number triangle defined by sequence A000217. Image under a real-rootedness preserver (reversal, shift 0)
- theta_plus_one of Stirling-like number triangle defined by sequence A000217. Image under a real-rootedness preserver (euler polynomial, shift 0)
- one_plus_derivative of Stirling-like number triangle defined by sequence A00021… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- shift_x_plus_one of Stirling-like number triangle defined by sequence A000217. Image under a real-rootedness preserver (affine, shift 0)
- mobius_x_over_x_plus_one of Stirling-like number triangle defined by sequence A… Image under a real-rootedness preserver (mobius, shift 0)
- theta_plus_one_squared of Stirling-like number triangle defined by sequence A00… Image under a real-rootedness preserver (euler polynomial, shift 0)
Related ?
Same fingerprint ?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.