Real-rooted atlas

A038284 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 46\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 896\,x + 1024\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
019689610242116
1274418816430083276897336
2384163512321204224183500810485764477456
3537824614656028098560642252807340032033554432205962976
4752953610326220859006976017983078403082813440281857228810737418249474296896
51054135041686616064115653672964405854208010070523904013811004211210522669875234359738368435817657216
61475789056269858570242158868561929869113425922819746693120515610823884858926951301123848290697216109951162777620047612231936
7206610467844250272481283885963411456207251381944327105761666662416241740952371224749319546470424244231392460813853846509977635184372088832922190162669056
82892546549766611534970880680043597004804145027638886401658011055554560454768746666393686622618412646401131397464981504096976925569843204925812092436480112589990684262442420747482776576
940495651696641018176385515521163630154874880797917820485632036476243222200320116723978311041024266797664710950912435588024017879040497814884591861760379287531117608960173388585653764096360287970189639681951354384207722496
105669391237529615550330251509761954898660189798414894465982398464076600110766620672028013754794649845767470334611906625536146357576070007357442090822515285819392021240101742586101760145646411949161840646052837899185946624115292150460684697689762301673555234816
117937147732541442358466754812313632344686923140300827107928087965204481549024462169440256063731292157828399104194228699909572263936443951314079022317568761059395564038258688966424629287667630080883588232491581833216550808248825921142784209831713838446149632368934881474191032324129065876983540801536
12111120068255580163555842184178565125282965530779582464483013991385561825283036087945852102901761387925918103818469376475860314778452046643212430636794212624891904248612735884252497838083788384546807657109913643295823392087509827584359861389232935146618882056350795616772266393672311236768941442334721180591620717411303424189937030341242876870656
131555680955578122245333763276267847680853402124202855628808452744849247331942405796167896626741903360291464442801801878568961110340734483054775500803263042158480814034124807458382076527574935142401325934591382679988469760181842458246767541275852818892722934729095197491201439445556931740586475520759267986073885144514560247924240350656373719040377789318629571617095688737103395697172336050176
14217795333780937113679650864925599858688136544339872456900608014564729586395402731520108195134070365848862720593527592614578370904064248716324524204269712179281213493722189149293772802088346981427720981839872042429906924245759631032320678878510787932154096517128463939874758634648476057680608951188177472842629120566920096268500907904204802776751491927351385653248084624807373024042229432321208925819614629174706176401906756202069927458308096
Technical details
Node id
n-da75ead823e98604 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5dbcc0e32705bbae

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:           196,             896,             1024;
  1:          2744,           18816,            43008,            32768;
  2:         38416,          351232,          1204224,          1835008,           1048576;
  3:        537824,         6146560,         28098560,         64225280,          73400320,           33554432;
  4:       7529536,       103262208,        590069760,       1798307840,        3082813440,         2818572288,         1073741824;
  5:     105413504,      1686616064,      11565367296,      44058542080,      100705239040,       138110042112,       105226698752,        34359738368;
  6:    1475789056,     26985857024,     215886856192,     986911342592,     2819746693120,      5156108238848,      5892695130112,      3848290697216,      1099511627776;
  7:   20661046784,    425027248128,    3885963411456,   20725138194432,    71057616666624,    162417409523712,    247493195464704,    242442313924608,    138538465099776,     35184372088832;
  8:  289254654976,   6611534970880,   68004359700480,  414502763888640,  1658011055554560,   4547687466663936,   8662261841264640,  11313974649815040,   9697692556984320,   4925812092436480,   1125899906842624;
  9: 4049565169664, 101817638551552, 1163630154874880, 7979178204856320, 36476243222200320, 116723978311041024, 266797664710950912, 435588024017879040, 497814884591861760, 379287531117608960, 173388585653764096, 36028797018963968;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (14+46*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 896*x + 1024*x^2.
%t Mathematica
P[0, x_] := 196 + 896*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14 + 46*x + 32*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] = 196 + 896*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14 + 46*x + 32*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.