Real-rooted atlas

A038337 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(34 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1156 + 748\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
011567481212025
1393043814812342133191125
213363361729376839256181016146414100625
3454354247349848047557840153863602488970161051184528125
415448044162998737984242544984010462724802538749403285440417715618303765625
5525233501441189499400321154514123846225321256020140745240390967407642163151819487171373669453125
61785793904896462205481267252337973614083386574763264136957067632035447711622457341886448530051051221435888116815125390625
76071699276646417679359658470422879171322726417271531292646483817725390784271174993911365848872417696810978108336655938175862357947691756680642578125
820643777540597766678869204311040972364781215872083890294849996804749671105477760184398995859724849715415550416091910852278080111509489896208017022149402593742460134050628916015625
970188843638032384249789708241232896404071586860817920392187128423735040253768141921240320114942040752561792371871308317111688593664688000480139015164070596014991831419378097005968007742853116706111532278301220703125
102386420683693101056926492736022027468816486120743921371136177791498218759884801294217523798325632066994789467207444482528724896556359424701243038540839168141795467352007920203888907303540801978921747357896116407161609288313842837672168952523554931640625
118113830324556543590434125815776811345100866244230625574964019278583842212691869081663560460613206658816037014621180632113075215967091489684440934451658237172508485376125347193139175001282252972425704125840291561137444063344257259827156526481387185342510682345227121439313102863559971923828125
12275870231034922482073612495298699817077129216262768781481447357276163400537172112848152985630254779251886369596416195766218688676509153289500419436362242355968351276012773057700556899442106557078834348821448297492703277996834695775355843537936408185592421688681633015011151754231616432810980511156379749833583241139628860198736572265625
13937958785518736439050244551858812076220954214401030862142734908863160320144522829814796046501888014027215834965498631065609984077153122501966817285383571013938604001715202239384581428242841049607245067763444315073984018231052868797786297280353896908629604086947252043663033765306904056125518957982193720419036680003034478019367241512745291041772481694156516283298708943145751953125
143189059870763703892770816165080746251297613272842244005635754627074439708672060477245707114653305405440635900451185102604608307204937579973907855518135091229286626315826005769330688135358356921884900614553604926646079142134250309120141681325151799939224576032086653049084103883212856623505380736653911552763307057828557834592075985317973883585344052678896914667191520227242300416211414445949729863572161282748441902441558837890625
Technical details
Node id
n-cef993e093439389 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bb98156118eead7a

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:              1156,                748,                121;
  1:             39304,              38148,              12342,               1331;
  2:           1336336,            1729376,             839256,             181016,              14641;
  3:          45435424,           73498480,           47557840,           15386360,            2488970,             161051;
  4:        1544804416,         2998737984,         2425449840,         1046272480,          253874940,           32854404,           1771561;
  5:       52523350144,       118949940032,       115451412384,        62253212560,        20140745240,         3909674076,         421631518,         19487171;
  6:     1785793904896,      4622054812672,      5233797361408,      3386574763264,      1369570676320,       354477116224,       57341886448,       5300510512,        214358881;
  7:    60716992766464,    176793596584704,    228791713227264,    172715312926464,     83817725390784,     27117499391136,     5848872417696,     810978108336,      65593817586,      2357947691;
  8:  2064377754059776,   6678869204311040,   9723647812158720,   8389029484999680,   4749671105477760,   1843989958597248,   497154155504160,   91910852278080,   11150948989620,    801702214940,   25937424601;
  9: 70188843638032384, 249789708241232896, 404071586860817920, 392187128423735040, 253768141921240320, 114942040752561792, 37187130831711168, 8593664688000480, 1390151640705960, 149918314193780, 9700596800774, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (34+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1156 + 748*x + 121*x^2.
%t Mathematica
P[0, x_] := 1156 + 748*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(34 + 11*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] = 1156 + 748*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (34 + 11*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

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.