A118800: Triangle read by rows: T satisfies the matrix products: C*T*C = T^-1 and T*C*T = C^-1, where C is Pascal's triangle.
A118800 ↗ only rational roots
Proved for all n all roots real, in [0, ∞)alternating coefficient signsinterlacing (weak)
Recurrence from A118800
\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 + 2\,x + 2\,n - n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 - x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 1 | -1 | 0 | |||||||||||||
| 3 | 2 | -3 | 1 | 0 | ||||||||||||
| 4 | 4 | -8 | 5 | -1 | 0 | |||||||||||
| 5 | 8 | -20 | 18 | -7 | 1 | 0 | ||||||||||
| 6 | 16 | -48 | 56 | -32 | 9 | -1 | 0 | |||||||||
| 7 | 32 | -112 | 160 | -120 | 50 | -11 | 1 | 0 | ||||||||
| 8 | 64 | -256 | 432 | -400 | 220 | -72 | 13 | -1 | 0 | |||||||
| 9 | 128 | -576 | 1120 | -1232 | 840 | -364 | 98 | -15 | 1 | 0 | ||||||
| 10 | 256 | -1280 | 2816 | -3584 | 2912 | -1568 | 560 | -128 | 17 | -1 | 0 | |||||
| 11 | 512 | -2816 | 6912 | -9984 | 9408 | -6048 | 2688 | -816 | 162 | -19 | 1 | 0 | ||||
| 12 | 1024 | -6144 | 16640 | -26880 | 28800 | -21504 | 11424 | -4320 | 1140 | -200 | 21 | -1 | 0 | |||
| 13 | 2048 | -13312 | 39424 | -70400 | 84480 | -71808 | 44352 | -20064 | 6600 | -1540 | 242 | -23 | 1 | 0 | ||
| 14 | 4096 | -28672 | 92160 | -180224 | 239360 | -228096 | 160512 | -84480 | 33264 | -9680 | 2024 | -288 | 25 | -1 | 0 | |
| 15 | 8192 | -61440 | 212992 | -452608 | 658944 | -695552 | 549120 | -329472 | 151008 | -52624 | 13728 | -2600 | 338 | -27 | 1 | 0 |
b-files: A118800 ?
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Technical details
- Node id
n-7c1541c019618244· revision 0 · JSON- Definition source
oeis_catalog:A118800- Fingerprint
f6-4d86f3a82b76e06e
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 -1)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): no - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes
Matches ?
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1, -1; 3: 2, -3, 1; 4: 4, -8, 5, -1; 5: 8, -20, 18, -7, 1; 6: 16, -48, 56, -32, 9, -1; 7: 32, -112, 160, -120, 50, -11, 1; 8: 64, -256, 432, -400, 220, -72, 13, -1; 9: 128, -576, 1120, -1232, 840, -364, 98, -15, 1; 10: 256, -1280, 2816, -3584, 2912, -1568, 560, -128, 17, -1;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (2*n-4-(n-2)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 - x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 - x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 - (n - 2)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 - x; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 - (n - 2)*x)*v[i - 1])/(n - 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
A118800 triangle, offset 0, from P1
%e ExampleTriangle begins: 0: 1; 1: 1, -1; 2: 2, -3, 1; 3: 4, -8, 5, -1; 4: 8, -20, 18, -7, 1; 5: 16, -48, 56, -32, 9, -1; 6: 32, -112, 160, -120, 50, -11, 1; 7: 64, -256, 432, -400, 220, -72, 13, -1; 8: 128, -576, 1120, -1232, 840, -364, 98, -15, 1; 9: 256, -1280, 2816, -3584, 2912, -1568, 560, -128, 17, -1;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (2*n-2-(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 - x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 - x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 - (n - 1)*x)*P[n - 1, x])/(n - 1)]];
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; v[2] = 1 - x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 - (n - 1)*x)*v[i - 1])/(n - 1)); 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 ≥ 1, is zero or real-rooted with all roots in [0, +∞)
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n(-x) have one sign
derived from Iterated real-rootedness preservers
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
Proof chains ? JSON
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.