Real-rooted atlas

A038226 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 54\,x + 43\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 946\,x + 1849\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
012194618492916
11331156096101779507157464
2146412289321342374349830834188018503056
316105131478152461019096203470188034055147008443459165024
4177156141551158406068135211647634062051238159702557238632136304924794911296
5194871715332398616253449279407421695451592648445853735484536634867449547732718186111071338925209984
621435888167035868249171725609271706218399235038265808701095742130744821416778010012239200377774161168820027760172301961339136
723579476918295688694712971440504441183152603586869375766301226271196177359338706753674330396118404186998209211571318274824995025926119368433904305912313344
825937424601101391750713017835730693605185923980563640127188904885581059663159019054361943572604408589043414868566010040636422505115374455528518731305273021611482313284249210832519264920576
928531167061112268401836273239791490436245281210020602505521985510701650430120320704021759626470344570266878538131329977412180371025669041039653436153344753832439690165261498935990739412992929373947122270711384956040305711104
103138428376721147220822035276316524767375843441244136355034140362760926577232095226890470441032437610347580545871327836346711140368156179448470783543085633929514716916862734636726017258929775388801251412266677361020139732439959630797262576401614787626176508399616
113452271214393117543814625870394114821975885964258979114987698820257638680556160210654055667159133454822121138628829422855436482632821787743889433224226440933224913038375261297778427632629545822675652626866192983487706743131293999086665714227204008548425343171826412428229078524333198531813531453579264
123797498335832412078267271064646252806882023869873982570760982778348288876356805648672440169396971389616893622640691860496638996715071817922079330365567530462181198395277276798483162047971575571084989986316730126261343484298609450227948073975861978188439995494708658228751411264612675139472780927422738853573441385037654491792720717930698493280256
134177248169415651244938642661190445670241194918348399511353479635132022888513314535208473008660151145050027928678744772974601744243838160644295374946428861887898303145146569967646010723882138544563192183282048372246151045209416662433498185409718571902858051504306600255241997522089762025813276052183054569900650442651231512191083961782853121299085317707036579795566191430796806918768257718637133824
144594972986357216128739467405579678888425889307544951308015370864736794123295201952798497242707936822018320800446858859916366413129906986915516273395927332285719965807789039280322453928821223592540704701120445974893948644787903203065947622209804928010535126537309806034129515923289761064789097194952913578414620128072534767504825968872248010728153886368910746743194805591643843804401964969180321366140257293120934623152015227573613485916806405226496
Technical details
Node id
n-fbbb220cb64580dd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d0faa4e84707c93d

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:          121,            946,            1849;
  1:         1331,          15609,           61017,            79507;
  2:        14641,         228932,         1342374,          3498308,           3418801;
  3:       161051,        3147815,        24610190,         96203470,         188034055,          147008443;
  4:      1771561,       41551158,       406068135,       2116476340,        6205123815,         9702557238,         6321363049;
  5:     19487171,      533239861,      6253449279,      40742169545,      159264844585,       373548453663,       486744954773,        271818611107;
  6:    214358881,     6703586824,     91717256092,     717062183992,     3503826580870,     10957421307448,     21416778010012,      23920037777416,      11688200277601;
  7:   2357947691,    82956886947,   1297144050444,   11831526035868,    69375766301226,    271196177359338,    706753674330396,    1184041869982092,    1157131827482499,     502592611936843;
  8:  25937424601,  1013917507130,  17835730693605,  185923980563640,  1271889048855810,   5966315901905436,  19435726044085890,   43414868566010040,   63642250511537445,   55285187313052730,   21611482313284249;
  9: 285311670611, 12268401836273, 239791490436245, 2812100206025055, 21985510701650430, 120320704021759626, 470344570266878538, 1313299774121803710, 2566904103965343615, 3344753832439690165, 2614989359907394129, 929293739471222707;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+54*x+43*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 946*x + 1849*x^2.
%t Mathematica
P[0, x_] := 121 + 946*x + 1849*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 54*x + 43*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] = 121 + 946*x + 1849*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 54*x + 43*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

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.