Real-rooted atlas

A075505 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(10 + 11\,x - 9\,x^{2} - 10\,x^{3} - 10\,n x - 20\,n x^{2} - 10\,n x^{3}\right)\,P_{n-1}(x) + \left(10\,x + 30\,x^{2} + 30\,x^{3} + 10\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 100 + 230\,x + 131\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0100230131461
1100037004460176110921
210000550001075009010027601290201
31000008100002330000313650020231505066518906301
41000000123000004951000095390000971240005061821010674211316616421
5100000001970000001068600000275501000039250900003186236600138624348025168688112779866961
610000000033500000002367500000078077000000145579510000163400686000109920044200409710707606524202561571597513521
710000000006010000000053810000000022020050000005155782500000750156577000069366439600003981587335000129772337045018399172545127858499660901
810000000000112300000000012507100000000622863000000001782840300000003231294287000003853088333700003028667583600001515726119755004388301885155056059305660511466577011823101
91000000000002157000000000029672600000000017775341000000006107330800000000134231684520000001982900025160000020017443068270000136753864193550006063954584615500157855483824666018342810431616182974196618403321
101000000000000421500000000000718675000000000051437530000000000209337370100000000547148093920000000973940030064000000121241434362520000010600499309610200006398963073114050002545597609803381006016631948854198064092522397888815022968188690293961
1110000000000000832100000000000017796900000000000015152504500000000007227754535000000000221287242023000000004650447965120000000069406121246320000000745529366350330000005743146002889252000031032214288506146000111880073684936547002420469034793952310237948848526923611323831666289066196621
12100000000000000165230000000000000451399100000000000045551947000000000000252411360200000000000894315980273000000000218436990759030000000038286962431728000000004911979267741690000000463544913014640500000031892158017464697800001558911840432535590000513683290085888545000102429444633964441050934609729435670605122131039631317410062101
13100000000000000032917000000000000001173956600000000000001398852781000000000000893580474700000000000036280087996580000000000101729594083404000000000206147932060941300000000309444200164428900000000347807533292536890000000292502887765076930800000181731286268583375180000810711731206674583400002459161223531258254200045465060394850767852003869662451082182032011596695125160124731850401
141000000000000000065695000000000000000313000750000000000000043871102900000000000000321019431851000000000000148173046600180000000000047232096318016600000000001093096347763262800000000018900462479353182300000000247872469678313919000000002477433250764245187800000018781978508347484400400000106343423051881764337800004361762747387839841020000122542579277556025516200021113970607845908838536016836505067572362863361121197517342359206459610721
Technical details
Node id
n-8857fecd171cfa47 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5989189fb55e3737

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 2)
  • 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:          100,            230,             131;
  1:         1000,           3700,            4460,             1761;
  2:        10000,          55000,          107500,            90100,            27601;
  3:       100000,         810000,         2330000,          3136500,          2023150,            506651;
  4:      1000000,       12300000,        49510000,         95390000,         97124000,          50618210,          10674211;
  5:     10000000,      197000000,      1068600000,       2755010000,       3925090000,        3186236600,        1386243480,         251686881;
  6:    100000000,     3350000000,     23675000000,      78077000000,     145579510000,      163400686000,      109920044200,       40971070760,        6524202561;
  7:   1000000000,    60100000000,    538100000000,    2202005000000,    5155782500000,     7501565770000,     6936643960000,     3981587335000,     1297723370450,     183991725451;
  8:  10000000000,  1123000000000,  12507100000000,   62286300000000,  178284030000000,   323129428700000,   385308833370000,   302866758360000,   151572611975500,   43883018851550,    5605930566051;
  9: 100000000000, 21570000000000, 296726000000000, 1777534100000000, 6107330800000000, 13423168452000000, 19829000251600000, 20017443068270000, 13675386419355000, 6063954584615500, 1578554838246660, 183428104316161;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (10-(10*n-11)*x-(20*n+9)*x^2-(10*n+10)*x^3)*P(n-1,x) + (10*x+30*x^2+30*x^3+10*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 100 + 230*x + 131*x^2.
%t Mathematica
P[0, x_] := 100 + 230*x + 131*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((10 - (10*n - 11)*x - (20*n + 9)*x^2 - (10*n + 10)*x^3)*P[n - 1, x] + (10*x + 30*x^2 + 30*x^3 + 10*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 100 + 230*x + 131*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((10 - (10*n - 11)*x - (20*n + 9)*x^2 - (10*n + 10)*x^3)*v[i - 1] + (10*x + 30*x^2 + 30*x^3 + 10*x^4)*deriv(v[i - 1], x))/(1 + x)); 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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.