Real-rooted atlas

A038301 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(20 + 51\,x + 31\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1240\,x + 961\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
040012409612601
18000372005766029791132651
2160000992000230640023832809235216765201
3320000024800000768800001191640009235210028629151345025251
464000000595200000230640000047665600005541126000343549812088750368117596287801
51280000000138880000006457920000016682960000025858588000024048486840012425051534027512614111897410677851
6256000000003174400000001722112000000533854720000010343435200000128258596480009940041227200440201825776085289103744145767944570401
7512000000000714240000000044282880000000160156416000000372363667200000577163684160000596402473632000396181643198400153520386739380264396221606712334165173090451
8102400000000001587200000000001107072000000000457589760000000012412122240000000230865473664000002982012368160000026412109546560000153520386739380005287924432134200819628286980801119042423827613001
9204800000000000349184000000000027061760000000000125837184000000000390095270400000000846506736768000000131208544199040000014526660250608000001125816169422120000581671687534762000180318223135776220254084768964048316071163615208263051
104096000000000000761856000000000006494822400000000003355658240000000000117028581120000000002902308811776000000052483417679616000000697279692029184000006754897016532720000046533735002780960000216381867762931464006098034455137159440787662783788549761309629344375621415601
118192000000000000016506880000000000001535139840000000000087247114240000000000338082567680000000000943250363827200000000194938408524288000000030215453321264640000003512546448597014400000302469277518076240000018753095206120726880007927444791678307272002047923237850229378602441754629744504259115791096563156692195651
1216384000000000000003555328000000000000035819929600000000000022208356352000000000009466311895040000000000293455668746240000000006822844298350080000000012086181328505856000000016391883426786067200000016938279541012269440000013127166644284508816000073989484722330867872000286709253299032113004006836912963284611925480756943935220796320321805345924720991301978201
1332768000000000000000761856000000000000000826613760000000000000055520890880000000000000258172142592000000000000880367006238720000000000227428143278336000000000045323179981896960000000007025092897194028800000000846913977050613472000000078762999865707052896000005549211354174815090400000286709253299032113004000010255369444926917888220002270831805662388960963002346526199184468592995141072642160770556400888251
14655360000000000000000162529280000000000000001889402880000000000000001366668083200000000000000688459046912000000000000025610676545126400000000000727770058490675200000000001611490843800780800000000002810037158877611520000000003871606752231375872000000004200693326171042821120000003551495266671881657856000002293674026392256904032000001093906074125537908076800003633330889059822337540800075088838373902994975843207274231217471852638284812094704750199298376445300801
Technical details
Node id
n-50ac85330e220de7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-74659e42bd6f64b0

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             400,             1240,               961;
  1:            8000,            37200,             57660,              29791;
  2:          160000,           992000,           2306400,            2383280,             923521;
  3:         3200000,         24800000,          76880000,          119164000,           92352100,           28629151;
  4:        64000000,        595200000,        2306400000,         4766560000,         5541126000,         3435498120,           887503681;
  5:      1280000000,      13888000000,       64579200000,       166829600000,       258585880000,       240484868400,        124250515340,         27512614111;
  6:     25600000000,     317440000000,     1722112000000,      5338547200000,     10343435200000,     12825859648000,       9940041227200,       4402018257760,        852891037441;
  7:    512000000000,    7142400000000,    44282880000000,    160156416000000,    372363667200000,    577163684160000,     596402473632000,     396181643198400,     153520386739380,     26439622160671;
  8:  10240000000000,  158720000000000,  1107072000000000,   4575897600000000,  12412122240000000,  23086547366400000,   29820123681600000,   26412109546560000,   15352038673938000,   5287924432134200,    819628286980801;
  9: 204800000000000, 3491840000000000, 27061760000000000, 125837184000000000, 390095270400000000, 846506736768000000, 1312085441990400000, 1452666025060800000, 1125816169422120000, 581671687534762000, 180318223135776220, 25408476896404831;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (20+51*x+31*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1240*x + 961*x^2.
%t Mathematica
P[0, x_] := 400 + 1240*x + 961*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((20 + 51*x + 31*x^2)*P[n - 1, 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] = 400 + 1240*x + 961*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((20 + 51*x + 31*x^2)*v[i - 1])/(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

Same fingerprint ?

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.