Real-rooted atlas

A038314 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(17 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(506 + 408\,x + 72\,x^{2} + 132\,n + 72\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1012 + 816\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
010128161441972
1280723643214688172880920
27453601347456874368235008207363222928
3192089924472160040423680174873603525120248832125615584
4484441408138304742416099776009701683203147724805076172829859844816154944
51201826982440693078272580879918084507937280020373534720528817766471066419235831808182286921088
629433423078411537539031041953267757056185881573785610819049472003911718666248461084262497462517764299816966828035518720
771338635596803178809692467262302710767616703176392540165018802492211223369146859520704109359923213054244290561315743989765159780352253582734495232
81714322289264648560636271616001907285815480320249210843070464021095291776204801204512598130688467382937190400120704461701120195813664358401754325319680619173642249350416886035456
940902281098792962262905421829324856500199392665600839205758811340808223957821325312055691570289180672264992771588751368813506815590400199162361806848028719337439232023157094219776743008370688342686707173922816
109699683803428044858899284782261862416292919037171138562712009570847948800302114073172082688023684998525416898561336597686940336128545127987268288512158643122680627200318659778890956804135584591249408303147415240704891610044825612493614167569764352
112288190468326760448151315067333477498884594144213016425267284723178993289920512105768373263070003200942595908296897986566158099616608393625629787034166098919424106299957517316259842749814126464204800497109255069892608586501087486279683940916398129152106993205379072453419929901689200640
125373391099778572288038441599867889575526412710465656012109905922572720759289198149632355837351771817666150435538173416391521075202639268543231314362368147794390798601447014462552771748807730790419842658736565701836846196877324598640640759221407743108710482110152248079155250928765760438272128391846454886416389937255604218052608
1312568078972324197498889672103979601430118400345974398811006179737607626279393607265943552011577243416801391673344012810144663785435981414410661452024917456322560067866905397376655032320332537379296853255782401251055434976154615808035716785725818263306247559489016752504832001138832111614663065601136909800358019072065479841691992064015407021574586368590362294414734111637504
142929233455529817912115224130711626862459197849692852198204173729136640022142361523904395503206403660614108931487652904960444566147205173440256409640992462924113395140526082924283983977359448473600162880572953703972077568070941307583328694566912024020264351542168623513662342025994155514134528120951824268040077312001681967426384733143040155919058334814044160838141973657498419218488425889503641621197868552723443522535424
Technical details
Node id
n-84d302b3dc0e5d17 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-efae2a5576a476b8

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:             1012,               816,               144;
  1:            28072,             36432,             14688,              1728;
  2:           745360,           1347456,            874368,            235008,             20736;
  3:         19208992,          44721600,          40423680,          17487360,           3525120,            248832;
  4:        484441408,        1383047424,        1609977600,         970168320,         314772480,          50761728,           2985984;
  5:      12018269824,       40693078272,       58087991808,       45079372800,       20373534720,        5288177664,         710664192,         35831808;
  6:     294334230784,     1153753903104,     1953267757056,     1858815737856,     1081904947200,      391171866624,       84610842624,       9746251776,        429981696;
  7:    7133863559680,    31788096924672,    62302710767616,    70317639254016,    50188024922112,    23369146859520,     7041093599232,    1305424429056,     131574398976,      5159780352;
  8:  171432228926464,   856063627161600,  1907285815480320,  2492108430704640,  2109529177620480,  1204512598130688,   467382937190400,  120704461701120,   19581366435840,   1754325319680,    61917364224;
  9: 4090228109879296, 22629054218293248, 56500199392665600, 83920575881134080, 82239578213253120, 55691570289180672, 26499277158875136, 8813506815590400, 1991623618068480, 287193374392320, 23157094219776, 743008370688;
%F Formula
The row polynomials satisfy (6*n+17+6*x)*P(n,x) = (132*n+506+(72*n+408)*x+72*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1012 + 816*x + 144*x^2.
%t Mathematica
P[0, x_] := 1012 + 816*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((132*n + 506 + (72*n + 408)*x + 72*x^2)*P[n - 1, x])/(6*n + 17 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1012 + 816*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 506 + (72*n + 408)*x + 72*x^2)*v[i - 1])/(6*n + 17 + 6*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.