Real-rooted atlas

mobius_x_over_x_plus_one of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 7\,x + 19\,x^{2} + 25\,x^{3} + 16\,x^{4} + 4\,x^{5} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 14\,x^{3} + 16\,x^{4} + 9\,x^{5} + 2\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28x29Pn(1)
0123
11612827
21125411612048351
31221726801520195213443846075
4140490302011000252323696033600172803840131463
517413441192063080217392507808813248881280618240253440460803413907
611403654445483219441532832507640012018848205349762516236821584640123110404193280645120103429791
712709972162456154324896626884255507213676198432717491258718873678840115278072422455351296026578944077414400103219203581223435
81528274745872927149096570013923207005761330100928417543206410071510272187889433602708304384029898086400248126054401497968640062086348801579253760185794560139498558263
91104276600212304032534280323011824225901084811695604160462481939201424694233603459588746246659056609281015986616320122179872768011439912960008165206425604290306048001563770880003530096640037158912006037616347587
1012068216238771157214689092017867034241522978987296144784032465675722688177128857664053727112778241311923102259225912057746432414203484794885338913310720055026091253760447005144678402797505593344013008406118400423054213120085837086720081749606400287444492409231
11141186176762821834466202312097463361609977950963275486464486443881371170562012236096640074110069800448221903307604992544390638598144109877884223897618261366611558402493119442468864277950454087680025048795358822401796143764602880100051435782144041716824145920012246091038720022562891366400196199055360014929010774254395
1218216178300210415178829898310005278677518464178060918457418947780483944986262131221424697026304093851474610150433636956673315849961821300640768245369606241259525044641047427072086657590678364160124188656545873920147845153215119360145132051630325760116147488655278080745178645436825603739012812767232014128130477260800377912080465920063764692992000051011754393600839982382565841063
1311641051928803879148001356308772028515591249640810992040004273459448544034288906272192021754342696486401115819788270387247015478095738880164627299756774400482982946946734080119374813032548352024935975028357447684407010209727856640658337855349178368082869049896733900808741317263857418240766188598117662720055126214899531776003199263815997849600146013920286474240050437872156672000012390755142205440019282443160780800142832912302080050897213545996785267
141327961523400614578263406189285900015411959559362578428181201631339797788457629088245905929602132699675163200012652336701691315261808231882719795225186614081689016328643257898144092160251509661590832640006235099228512408780813205971364622613708823923064401333513420837045849742589542400048926728050226790400054880593518808465408051948400979533234176041116548157495050240026863960795982069760014230100848803840000059554837435200307200189450004054671360004302841483100160000621323168514048000428498736906240003304312091004451756671
Technical details
Node id
n-b38b95ff2228e17d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8569cda4d5c26e17

Observed on the stored rows ?

  • Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1,    2;
  1: 1,    6,    12,       8;
  2: 1,   12,    54,     116,      120,        48;
  3: 1,   22,   172,     680,     1520,      1952,       1344,         384;
  4: 1,   40,   490,    3020,    11000,     25232,      36960,       33600,       17280,         3840;
  5: 1,   74,  1344,   11920,    63080,    217392,     507808,      813248,      881280,       618240,       253440,        46080;
  6: 1,  140,  3654,   44548,   321944,   1532832,    5076400,    12018848,    20534976,     25162368,     21584640,     12311040,       4193280,        645120;
  7: 1,  270,  9972,  162456,  1543248,   9662688,   42555072,   136761984,   327174912,    587188736,    788401152,    780724224,     553512960,     265789440,      77414400,     10321920;
  8: 1,  528, 27474,  587292,  7149096,  57001392,  320700576,  1330100928,  4175432064,  10071510272,  18788943360,  27083043840,   29898086400,   24812605440,   14979686400,   6208634880,   1579253760,    185794560;
  9: 1, 1042, 76600, 2123040, 32534280, 323011824, 2259010848, 11695604160, 46248193920, 142469423360, 345958874624, 665905660928, 1015986616320, 1221798727680, 1143991296000, 816520642560, 429030604800, 156377088000, 35300966400, 3715891200;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-7)*x-(8*n-19)*x^2-(12*n-25)*x^3-(8*n-16)*x^4-(2*n-4)*x^5)*P(n-1,x) + (x+6*x^2+14*x^3+16*x^4+9*x^5+2*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (2*n - 7)*x - (8*n - 19)*x^2 - (12*n - 25)*x^3 - (8*n - 16)*x^4 - (2*n - 4)*x^5)*P[n - 1, x] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((1 - (2*n - 7)*x - (8*n - 19)*x^2 - (12*n - 25)*x^3 - (8*n - 16)*x^4 - (2*n - 4)*x^5)*v[i - 1] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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