Real-rooted atlas

A013612 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 32\,x + 26\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 312\,x + 676\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0363126761024
12162808121681757632768
21296224641460164218244569761048576
3777616848014601606327360137092801188137633554432
446656121305613141440759283202467670404277295363089157761073741824
527993684913921103880967972473603454738560898232025612974462592803181017634359738368
61679616582266888831047687653574656414568627201437171240963113871022083855268884482088270645761099511627776
710077696393030144681252249668882171904447734117376194018117529656049678397441040922598809611276661487104542950367897635184372088832
860466176262020096051093918720590418616320447734117376023282174103552840745175961602081845197619203382998446131203257702207385601411670956533761125899906842624
936279705617293326336374688737280487095358464042214931066880256103915139072110978363226931234350445760716807442596581488640107504172843724809317028313122816367034448698777636028797018963968
10217678233611319268147226977589084163896762867712037993437960192026342116985733121331740358723174449464641895432192133966738466795520258010014824939520335413019272421376264264803063119872954289566616821761152921504606846976
111306069401673575242956819129563168768303947503681536329276462321664025683564061089792148393925686296576643040344640618496208988112008201011250311952890863206408720738501082955776103063273194616750087443458619611209728248115287320373657636893488147419103232
12783641640964754092621824133906942181376232105366447718427659222835019776239713264570171392155813621970611404877164841356874219522925833568114814156884524080856650186752183135508522742071296288577164944926900224312625262023670808576208416841349113872384645099747032971509761180591620717411303424
13470184984576305620239974409270480612556801740790248357888022630273228652544021574193811315425281558136219706114048086810446526483496960376178601614761820160126786121284975280128032964391534093572833286492986211260855255040937875786071012425728093787578607101242572805805897723296743587840167725934228572592537637778931862957161709568
142821109907456195596953583616635690099146752012855066449412096018104218582922035201882838732623891660814958107709178694860892597809628249063424045141432193771418419201738781091908232413184052743026454549716533248124665335256208420896768225090188657042982174720300120251542723976232960278683090718243692216320161016896859429688836096436087428994288740597761208925819614629174706176
Technical details
Node id
n-2ab248536be3787d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a39bea8eb66cb9fc

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:        36,         312,          676;
  1:       216,        2808,        12168,         17576;
  2:      1296,       22464,       146016,        421824,         456976;
  3:      7776,      168480,      1460160,       6327360,       13709280,        11881376;
  4:     46656,     1213056,     13141440,      75928320,      246767040,       427729536,        308915776;
  5:    279936,     8491392,    110388096,     797247360,     3454738560,      8982320256,      12974462592,       8031810176;
  6:   1679616,    58226688,    883104768,    7653574656,    41456862720,    143717124096,     311387102208,     385526888448,     208827064576;
  7:  10077696,   393030144,   6812522496,   68882171904,   447734117376,   1940181175296,    5604967839744,   10409225988096,   11276661487104,     5429503678976;
  8:  60466176,  2620200960,  51093918720,  590418616320,  4477341173760,  23282174103552,   84074517596160,  208184519761920,  338299844613120,   325770220738560,  141167095653376;
  9: 362797056, 17293326336, 374688737280, 4870953584640, 42214931066880, 256103915139072, 1109783632269312, 3435044576071680, 7442596581488640, 10750417284372480, 9317028313122816, 3670344486987776;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (6+32*x+26*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 312*x + 676*x^2.
%t Mathematica
P[0, x_] := 36 + 312*x + 676*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 32*x + 26*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] = 36 + 312*x + 676*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 32*x + 26*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.