Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2*x - (2*t - 1))*Product_{t=1..n}(2*x + (2*t - 1)).
A380570 (tail) ↗ only rational roots
Proved for all n all roots real, in [0, ∞)alternating coefficient signsinterlacing (weak)
Recurrence from A380570
\(P_{n}(x) = \left(4 - x - 4\,n x - 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 16 - 40\,x + 9\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 16 | -40 | 9 | -15 | |||||||
| 2 | 64 | -560 | 1036 | -225 | 315 | ||||||
| 3 | 256 | -5376 | 31584 | -51664 | 11025 | -14175 | |||||
| 4 | 1024 | -42240 | 561792 | -2764960 | 4228884 | -893025 | 1091475 | ||||
| 5 | 4096 | -292864 | 7358208 | -79036672 | 351475696 | -515267064 | 108056025 | -127702575 | |||
| 6 | 16384 | -1863680 | 78926848 | -1559683840 | 14763100352 | -61460460880 | 87512357916 | -18261468225 | 21070924875 | ||
| 7 | 65536 | -11141120 | 735035392 | -23997276160 | 409981265408 | -3567539422720 | 14178653129664 | -19763326404000 | 4108830350625 | -4656674397375 | |
| 8 | 262144 | -63504384 | 6159925248 | -308414332928 | 8575137871872 | -132754743393792 | 1087733505684736 | -4176684060088896 | 5728036652158500 | -1187451971330625 | 1327152203251875 |
Compute more
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Technical details
- Node id
n-682c4fc0ca883d5a· revision 0 · JSON- Definition source
oeis_catalog:A380570- Fingerprint
f6-e501998dc2c96ec8
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 ?
Row sums ? A380612
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 16, -40, 9; 2: 64, -560, 1036, -225; 3: 256, -5376, 31584, -51664, 11025; 4: 1024, -42240, 561792, -2764960, 4228884, -893025; 5: 4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025; 6: 16384, -1863680, 78926848, -1559683840, 14763100352, -61460460880, 87512357916, -18261468225; 7: 65536, -11141120, 735035392, -23997276160, 409981265408, -3567539422720, 14178653129664, -19763326404000, 4108830350625; 8: 262144, -63504384, 6159925248, -308414332928, 8575137871872, -132754743393792, 1087733505684736, -4176684060088896, 5728036652158500, -1187451971330625; 9: 1048576, -348651520, 47564783616, -3457390346240, 145638125674496, -3626643745320960, 52275396387897856, -409378531792545280, 1530695092300725456, -2072571039314541000, 428670161650355625; 10: 4194304, -1857028096, 344014454784, -34805630959616, 2107261645389824, -78732988403736576, 1808451477238134784, -24690963934233135616, 186658712889715370304, -683326819861878090096, 915718508984314003500, -189043541287806830625;
%F FormulaThe row polynomials satisfy P(n,x) = (4-(4*n^2+4*n+1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 16 - 40*x + 9*x^2.
%t MathematicaP[1, x_] := 16 - 40*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(4 - (4*n^2 + 4*n + 1)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 16 - 40*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = (4 - (4*n^2 + 4*n + 1)*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, v[i])))};
T(10)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.