A028326 → D → D+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - x - 2\,n + n x + n^{2}\right)\,P_{n}(x) = \left(n x + n x^{2} + n^{2} + n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 2 | 2 | ||||||
| 2 | 8 | 4 | 12 | |||||
| 3 | 18 | 24 | 6 | 48 | ||||
| 4 | 32 | 72 | 48 | 8 | 160 | |||
| 5 | 50 | 160 | 180 | 80 | 10 | 480 | ||
| 6 | 72 | 300 | 480 | 360 | 120 | 12 | 1344 | |
| 7 | 98 | 504 | 1050 | 1120 | 630 | 168 | 14 | 3584 |
Compute more
?
Technical details
- Node id
n-ae892ec0e2e55006· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-e99c3824c4864868
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 2; 2: 8, 4; 3: 18, 24, 6; 4: 32, 72, 48, 8; 5: 50, 160, 180, 80, 10; 6: 72, 300, 480, 360, 120, 12; 7: 98, 504, 1050, 1120, 630, 168, 14; 8: 128, 784, 2016, 2800, 2240, 1008, 224, 16; 9: 162, 1152, 3528, 6048, 6300, 4032, 1512, 288, 18;
%F FormulaThe row polynomials satisfy (n^2-2*n+1+(n-1)*x)*P(n,x) = (n^2+(n^2+n)*x+n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + (n^2 + n)*x + n*x^2)*P[n - 1, x])/(n^2 - 2*n + 1 + (n - 1)*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; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + (n^2 + n)*x + n*x^2)*v[i - 1])/(n^2 - 2*n + 1 + (n - 1)*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]
-
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
-
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
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
-
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
-
from
Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.?
-
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
-
from
Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.?
-
-
from
derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
Related ?
Same fingerprint ?
- basis-map coefficients of A026729 → basis A007318 → D+1
- A085880 → basis A013609 → D+1
- basis-map coefficients of A026729 → basis A007318 → θ+1
- A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
- A131431 → x ↦ x+1 → θ+1
- A178821 → basis A013609 → θ+1
- A038325 → basis A013609 → θ+1
- A344565 → D+1
- A356546 → basis A013609 → D+1
- basis-map coefficients of A113214, shifted → D → D+1
- A038208 → D+1
- basis-map coefficients of A113214, shifted → basis A013609 → θ+1
- A007318 → ×(−(1/2)n+(1/2)n²) → D+1
- A196347 → θ+1
- A038208 → θ+1
- A156992 → basis A013609 → D+1
- A376827 → θ+1
- A038325 → D+1
- A038208 → D → θ+1
- A156992 → θ+1
- A038221 → basis A013609 → D+1
- A094305 → D+1
- A003506 → D → θ+1
- basis-map coefficients of A026729, shifted → basis A007318 → θ+1
- A038221 → θ+1
- A038338 → D+1
- A038338 → basis A013609 → D+1
- A093375 → D → D
- A038338 → basis A013609 → θ+1
- A124860 → θ+1
- basis-map coefficients of A113214, shifted → D → θ+1
- A135838 → D+1
- A120910 → basis A013610 → D+1
- A196347 → D+1
- A027465 → basis A013611 → D+1
- A131431 → x ↦ x+1 → D+1
- A124860 → D+1
- basis-map coefficients of A113214, shifted → basis A013609 → D+1
- A178821 → θ+1
- A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
- A028326 → D+1
- A085881 → basis A013609 → θ+1
- A156992 → D+1
- A356546 → basis A013609 → θ+1
- A216973 → θ+1
- A156992 → basis A013609 → θ+1
- basis-map coefficients of A113214, shifted → θ+1
- A178822 → basis A013609 → θ+1
- A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
- basis-map coefficients of A026729, shifted → basis A007318 → D+1
- A178820 → θ+1
- A093375 → D
- A027465 → basis A013611 → θ+1
- A376827 → D+1
- A038221 → D+1
- basis-map coefficients of A127647 → D+1
- A178820 → D+1
- A135065 → θ+1
- A178822 → θ+1
- basis-map coefficients of A127647 → θ+1
- A085880 → D → θ+1
- A178820 → basis A013609 → D+1
- basis-map coefficients of A112227 → x ↦ x+1 → D+1
- A085880 → D → D+1
- A196347 → basis A013609 → D+1
- A178822 → basis A013609 → D+1
- A085880 → basis A013609 → θ+1
- A134346 → θ+1
- A028326 → basis A013609 → θ+1
- A038208 → D → D+1
- A085881 → D+1
- A028326 → D → θ+1
- A356546 → θ+1
- A135838 → θ+1
- A120910 → basis A013610 → θ+1
- A178822 → D+1
- A196347 → basis A013609 → θ+1
- A356546 → D+1
- A178821 → D+1
- A038338 → θ+1
- A178820 → basis A013609 → θ+1
- A038330 → basis A013611 → D+1
- A038325 → basis A013609 → D+1
- A094305 → θ+1
- basis-map coefficients of A112227 → x ↦ x+1 → θ+1
- A085881 → θ+1
- A134346 → D+1
- A131431 → D+1 → basis A007318
- A085880 → D+1
- A135065 → D+1
- A085880 → θ+1
- A028326 → θ+1
- A178821 → basis A013609 → D+1
- A038221 → basis A013609 → θ+1
- basis-map coefficients of A113214, shifted → D+1
- A038330 → basis A013611 → θ+1
- A028326 → basis A013609 → D+1
- A038325 → θ+1
- A085881 → basis A013609 → D+1
- A344565 → θ+1
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
- [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.