Real-rooted atlas

A038320 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 17\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 816\,x + 289\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
05768162891681
1138242937620808491368921
2331776940032998784471648835212825761
37962624282009603995136028298880100225201419857115856201
419110297681218764814382489601358346240721621440204459408241375694750104241
5458647142422741254144483251650565705054208040410800640171745902724055111592410338673194754273881
611007531417662376011366415464052817922190740815872193971843072010991737774083892907128327878502521669757574417984925229121
7264180754022416841523068928477176486952967886666937139283795836207104593553839800322802893132390485087827233281506763607256118587876497327381934393961
8634033809653764491072818380801431529460858880270400009273344033518334482841602849058431041536168173587943424068070261786624018081163287072028461090359280201599390044913422659310152401
91521681143169024118564322405253124199153085186048089232003060203520126412004335288320125358570965827584887956544341278724492637277917184015911423692623360375686392742496053222238971853634271896307633550329031716248441
1036520347436056576310422953206480896120935608853358182428554240979265126404550832156070379520515760977687976345642621914128381378562587759072080297984114562250586888192036065893703279616076640024119469184987030613659830458262223722976122563490300366186081
11876488338465357824807099678336850329634301736329316139008890892318553071943681577621814104398233602011467812983107747841899719601150712872961345634717481754951687148684436621823180828131397088558100480797056250842479513615397677573093354241817781380156854329904578032905937925103102315013629321
1221035720123168587776208604224554755162112960448617220851892224272127108212574702796853008092953907780648967509479835136935591936797882232483299406643264590466439124237680644003263284508220981248189042988435110435225666952725070768279142417245398881864556748830538727186635152576332793821905639483216837782655940092937929227194915558802161
1350485728295604610662453641086314079898828802659703863073128316928081638132463772410839040173481031485516373032960270341274064929681309696319152892993319762657280290657098976059069562880205882111774708507607040113425793061066261135360482059620509531609825281552085899367810107392036646472623962183091205990288794301510697606061601756138433444028624230515098157931555098314991537910888601
141211657479094510655897613732118096404454100172872951877387148662407168024114648358529696740147205551393007536523937054720943736811281209069299302412255471090943478886039552124013695563118536346828809882341365186008365137920622221493363563489656832030851815712610023028817921192001970714478162477056351806137190036957675520766756965670593369292801163827537178579221248010991704517797692645124866119187566686848163759030914653054346432641
Technical details
Node id
n-70c59802becb9104 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-3d4815a526d230d8

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:              576,               816,               289;
  1:            13824,             29376,             20808,              4913;
  2:           331776,            940032,            998784,            471648,              83521;
  3:          7962624,          28200960,          39951360,          28298880,           10022520,            1419857;
  4:        191102976,         812187648,        1438248960,        1358346240,          721621440,          204459408,          24137569;
  5:       4586471424,       22741254144,       48325165056,       57050542080,        40410800640,        17174590272,        4055111592,         410338673;
  6:     110075314176,      623760113664,     1546405281792,     2190740815872,      1939718430720,      1099173777408,      389290712832,       78785025216,        6975757441;
  7:    2641807540224,    16841523068928,    47717648695296,    78866669371392,     83795836207104,     59355383980032,    28028931323904,     8508782723328,     1506763607256,     118587876497;
  8:   63403380965376,   449107281838080,  1431529460858880,  2704000092733440,   3351833448284160,   2849058431041536,  1681735879434240,   680702617866240,   180811632870720,   28461090359280,   2015993900449;
  9: 1521681143169024, 11856432240525312, 41991530851860480, 89232003060203520, 126412004335288320, 125358570965827584, 88795654434127872, 44926372779171840, 15911423692623360, 3756863927424960, 532222389718536, 34271896307633;
%F Formula
The row polynomials satisfy P(n,x) = (24+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 816*x + 289*x^2.
%t Mathematica
P[0, x_] := 576 + 816*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 17*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 576 + 816*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 17*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 - 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.