Real-rooted atlas

A013618 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 57\,x + 45\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 1080\,x + 2025\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144108020253249
11728194407290091125185193
22073631104017496004374000410062510556001
3248832466560034992000131220000246037500184528125601692057
42985984671846406298560003149280000885735000013286025000830376562534296447249
53583180894058496010581580800661348800002480058000005580130500006975163125003736694531251954897493193
6429981696128994508801693052928001269789696000595213920000017856417600000334807830000003587226750000016815125390625111429157112001
751597803521741425868802612138803200228562145280001285662067200004821232752000001205308188000000193710244500000018160335421875007566806425781256351461955384057
86191736422423219011584003918208204800039182082048000025713241344000001157095860480000036159245640000000774840978000000001089620125312500009080167710937500034050628916015625362033331456891249
9743008370688306490952908805746705367040006465043537920000484878265344000002545610893056000009546040848960000002556975227400000000479432855137500000059929106892187500004494683016914062500153227830122070312520635899893042801193
108916100448256401224520171520827525572853760010344069660672000087278087761920000052366852657152000002291049803750400000073640886549120000000172595827849500000000287659713082500000000323617177217812500000220648075375781250000689525235549316406251176246293903439668001
1110699320537907252159187622297601173581721501696001613674867064832000151282018787328000001021153626814464000005105768134072320000001914663050277120000000538498982890440000000011218728810217500000000168280932153262500000001721054987931093750000010756593674569335937500310286355997192382812567046038752496061076057
12128391846454886467405719388815360164301441010237440024645216151535616000254153791562711040000190615343672033280000010722113081551872000000459519132066508800000001507797152093232000000003769492880233080000000007067799150437025000000009637907932414125000000009035538686638242187500005212810780752832031250001396288601987365722656253821624208892275481335249
131540702157458636886664496357048320022749430293725184000369678242273034240000415888022557163520000034310761860965990400000214442261631037440000000103391804714964480000000038771926768111680000000001130847864069924000000000025444076941573290000000000433705856958635625000000005421323211982945312500000046915297026775488281250000251331948357725830078125006283298708943145751953125217832579906859702436109193
1418488425889503641611093055533702184960311992186885373952000545986327049404416000066542083609146163200000598878752482315468800000411729142331591884800000022056918339192422400000000930526242434680320000000003101754141448934400000000008142104621303452800000000001665430490721160800000000000260223514175181375000000000030025790097136312500000000002412786704234167968750000000120639335211708398437500000028274844190244155883789062512416457054691003038858224001
Technical details
Node id
n-6cb9a6f2f8e7a17f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1bc3b9efcd292160

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:          144,           1080,            2025;
  1:         1728,          19440,           72900,            91125;
  2:        20736,         311040,         1749600,          4374000,           4100625;
  3:       248832,        4665600,        34992000,        131220000,         246037500,          184528125;
  4:      2985984,       67184640,       629856000,       3149280000,        8857350000,        13286025000,         8303765625;
  5:     35831808,      940584960,     10581580800,      66134880000,      248005800000,       558013050000,       697516312500,        373669453125;
  6:    429981696,    12899450880,    169305292800,    1269789696000,     5952139200000,     17856417600000,     33480783000000,      35872267500000,      16815125390625;
  7:   5159780352,   174142586880,   2612138803200,   22856214528000,   128566206720000,    482123275200000,   1205308188000000,    1937102445000000,    1816033542187500,     756680642578125;
  8:  61917364224,  2321901158400,  39182082048000,  391820820480000,  2571324134400000,  11570958604800000,  36159245640000000,   77484097800000000,  108962012531250000,   90801677109375000,   34050628916015625;
  9: 743008370688, 30649095290880, 574670536704000, 6465043537920000, 48487826534400000, 254561089305600000, 954604084896000000, 2556975227400000000, 4794328551375000000, 5992910689218750000, 4494683016914062500, 1532278301220703125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (12+57*x+45*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1080*x + 2025*x^2.
%t Mathematica
P[0, x_] := 144 + 1080*x + 2025*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 57*x + 45*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] = 144 + 1080*x + 2025*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 57*x + 45*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.