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 | x10 | x11 | x12 | x13 | x14 | x15 | x16 | 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 | |||||||
| 9 | 1048576 | -348651520 | 47564783616 | -3457390346240 | 145638125674496 | -3626643745320960 | 52275396387897856 | -409378531792545280 | 1530695092300725456 | -2072571039314541000 | 428670161650355625 | -473793336560919375 | ||||||
| 10 | 4194304 | -1857028096 | 344014454784 | -34805630959616 | 2107261645389824 | -78732988403736576 | 1808451477238134784 | -24690963934233135616 | 186658712889715370304 | -683326819861878090096 | 915718508984314003500 | -189043541287806830625 | 207047688077121766875 | |||||
| 11 | 16777216 | -9646899200 | 2358425681920 | -321206170419200 | 26841225359196160 | -1429673364026163200 | 48883556774529187840 | -1055434687195905843200 | 13808154772768190222080 | -101475766398106943251200 | 365142761742870765674784 | -485171265417853335174000 | 100004033341249813400625 | -108700036240488927609375 | ||||
| 12 | 67108864 | -49073356800 | 15463014727680 | -2758840732876800 | 308118757948784640 | -22494459305602252800 | 1089080079614468751360 | -34773961732864365772800 | 714879298588513912888320 | -9035999798572546661804800 | 64882925045788322594699136 | -230154911150965641887436000 | 303632057019523333737352500 | -62502520838281133375390625 | 67502722505343624045421875 | |||
| 13 | 268435456 | -245215789056 | 97626536017920 | -22307900667985920 | 3243669926062325760 | -314596411767073013760 | 20754781152241917296640 | -933035224970405182832640 | 28209735297612178299924480 | -557291007865316829142804480 | 6846775553342539806834495744 | -48220272002983549739085414144 | 168997458457132046270890254000 | -221597779650585634828031535000 | 45564337691106946230659765625 | -48939473816374127432930859375 | ||
| 14 | 1073741824 | -1206617374720 | 596732622667776 | -171335519463014400 | 31735624166025461760 | -3986312054886708019200 | 347594706905076073758720 | -21186911848917073177804800 | 897521565390559471961948160 | -25953551416753109266807705600 | 496068839828101612536436550656 | -5951019328373010176504152577280 | 41229238588337693515654394311104 | -143013253681050393453130829754000 | 186545990036906946675297159997500 | -38319607998220941779984862890625 | 40962339584305144661363129296875 | |
| 15 | 4294967296 | -5858335391744 | 3546489787777024 | -1258802128235790336 | 291595930868058685440 | -46443183043097300828160 | 5221224712366430701486080 | -418786160731446399593349120 | 23950708548371545211718205440 | -966332430007340089622663004160 | 26925638270812144455547951284224 | -500526232388297690353532135489536 | 5883846528919813553683108204010496 | -40193351298116725042356396251986944 | 138181920747637055895159916033584000 | -179423974857460459522080510209160000 | 36825143286290325050565453237890625 | -39200958982180023440924514737109375 |
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.