Real-rooted atlas

A038330 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 44\,x + 28\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 896\,x + 784\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02568967841936
1409621504376322195285184
265536458752120422414049286146563748096
31048576917504032112640561971204917248017210368164916224
4167772161761607687707033601798307840236027904016521953284818903047256313856
52684354563288334336172637552645035261952088117084160925229383685397171404813492928512319277809664
64294967296601295421443682934456321289027059712281974669312039476453703683454189699072172709484953637780199833614048223625216
7687194767361082331758592757632231014430936649433088812087047618561421152333332481658011055554561243508291665925440348776038410578455953408618121839509504
810995116277761924145348608015152644620288070712341561344021655654603161604547687466663936663204422221824066320442222182404352279020830720169255295254528029619676669542427197360938418176
917592186044416338649581355008296318383685632015556715143495680544485030022348801333988323554754562334479566220820482918099457776025602553337025554022401489446598239846405213063093839462482935094674718721196683881290399744
10281474976710656591097451092377656893129667641344331876589727907840130676407205363712036589394017501839367470334611906625536112055019178599383041225601772265930752095324582287350169605004540570085883904159235381775459942423221826508921241652654090776777588736
1145035996273704961024568915226787841075797360988127232690303306634048307230200769665239613440951324244455047823362219756570395111587843884573998191445278725098503372626271928324956878278942208819203469814795259546173441656047970464783400964830139913855618252865021114224979476482316779994178213904384
127205759403792793617654110539292344322008155073844504166414057085516911529164867649724050136734105623677403417547856936966215318397106312445952124306367942126248919041903441259113808186572822206814689661095510016194309628534534585712641236515817947038272716854097567035182924431361456472958639540273152182059119829942534144101938319743841411792896
13115292150460684697630264189495929733120370736321325139230720281141710338230583296014759939792757105623040568257682021148566487041657418239228349985587203729191038263787467571206526084316961628068249608882725875864438204006409326862169657660114206727419094907682229636300804327805362814633954508801747767550367448327782404369418875918620819456050976553552383909560324485286068729022118887424
1418446744073709551616516508834063867445248677917844708826021888055363290651220791787520314878715578818253291520132249060543103666382438442429906924245759631032321060747673106143990775808020883469814277209818398720324853974888756597175091203979461192387268315394867237985765927333015737860096276979543220136573088563201491428309646889239707648055928561611758346489036801304999770941028084744192142734349946674946768896197352587024076973231046656
Technical details
Node id
n-8b06684614b7f078 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d36c4f1a2e9bd6b7

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:            256,             896,              784;
  1:           4096,           21504,            37632,             21952;
  2:          65536,          458752,          1204224,           1404928,            614656;
  3:        1048576,         9175040,         32112640,          56197120,          49172480,           17210368;
  4:       16777216,       176160768,        770703360,        1798307840,        2360279040,         1652195328,          481890304;
  5:      268435456,      3288334336,      17263755264,       50352619520,       88117084160,        92522938368,        53971714048,        13492928512;
  6:     4294967296,     60129542144,     368293445632,     1289027059712,     2819746693120,      3947645370368,      3454189699072,      1727094849536,       377801998336;
  7:    68719476736,   1082331758592,    7576322310144,    30936649433088,    81208704761856,    142115233333248,    165801105555456,    124350829166592,     54403487760384,     10578455953408;
  8:  1099511627776,  19241453486080,  151526446202880,   707123415613440,  2165565460316160,   4547687466663936,   6632044222218240,   6632044222218240,   4352279020830720,   1692552952545280,   296196766695424;
  9: 17592186044416, 338649581355008, 2963183836856320, 15556715143495680, 54448503002234880, 133398832355475456, 233447956622082048, 291809945777602560, 255333702555402240, 148944659823984640, 52130630938394624, 8293509467471872;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+44*x+28*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 896*x + 784*x^2.
%t Mathematica
P[0, x_] := 256 + 896*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + 44*x + 28*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] = 256 + 896*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + 44*x + 28*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.