A136124 → basis A013609 → x ↦ x+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(4 + 2\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 5 | 2 | 7 | ||||||
| 2 | 30 | 22 | 4 | 56 | |||||
| 3 | 210 | 214 | 72 | 8 | 504 | ||||
| 4 | 1680 | 2132 | 1004 | 208 | 16 | 5040 | |||
| 5 | 15120 | 22548 | 13300 | 3880 | 560 | 32 | 55440 | ||
| 6 | 151200 | 255720 | 178096 | 65400 | 13360 | 1440 | 64 | 665280 | |
| 7 | 1663200 | 3115320 | 2470496 | 1075592 | 277760 | 42560 | 3584 | 128 | 8648640 |
Compute more
?
Technical details
- Node id
n-d008e423f598159c· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-f01da5544602b4b6
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 5, 2; 2: 30, 22, 4; 3: 210, 214, 72, 8; 4: 1680, 2132, 1004, 208, 16; 5: 15120, 22548, 13300, 3880, 560, 32; 6: 151200, 255720, 178096, 65400, 13360, 1440, 64; 7: 1663200, 3115320, 2470496, 1075592, 277760, 42560, 3584, 128; 8: 19958400, 40710240, 35876592, 17848096, 5484304, 1066240, 128128, 8704, 256; 9: 259459200, 569149920, 547816176, 303778432, 106992144, 24829728, 3798144, 369408, 20736, 512;
%F FormulaThe row polynomials satisfy P(n,x) = (n+4+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 4 + 2*x)*P[n - 1, x]];
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] = (n + 4 + 2*x)*v[i - 1]); 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 (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
Proof chains ? JSON
-
Geometric basis map (elementary) polytool ? step ›
-
from
basis_map_A013609 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirli…?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…?
- ↑ Liu–Wang theorem (above)
- ↑ Iterated real-rootedness preservers (above)
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
- ↑ Liu–Wang theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…?
- ↑ Liu–Wang theorem (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…?
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Parent-row provers of polytool (above)
-
from
Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
- ↑ Liu–Wang theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
- ↑ Liu–Wang theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
- ↑ Liu–Wang theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…?
-
from
Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…?
- ↑ Liu–Wang theorem (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…?
-
-
from
basis_map_A013609 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirli…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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
- [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.