Real-rooted atlas

A038337 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(24 + 22\,x + 12\,n + 11\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 264 + 242\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0264242506
147528712399317457
27603220908819166458564535348
3114048041817605749920351384080525515391255
416422912752716801379980801264982405797836010629366424798638
5229920768126456422428979596803541950720243509112089286674413641019711398763453
631531991042023302758455640825856850068172807792291584042857603712130953789121714871048299624639336
7425681879043121667112961001534865408183614725324821039187276801542873733632707150461248185206073184212215292197752787542819
856757583872046825006694401716916912128036722945064960504940494643204628621200896028286018449920111123643910402546583506280259374246010198126792760930
9749200107110468676676485120283291290501120692489821224960111086908821504012219559970365449334386088473604889440332057601680745114144803423740047332031384283767215012607856851529
1098077104930816988944141385728453266064801792012464816782049280228521641004236802932694392887705626883031934803968176019851954073608067576547895040246509283407904045193368624782437661140520652125770888044638364
11127500236410060814025026005106688707095061090795522160568242221875204456171999582617606535718932721172486989588303049031685491819380967096323146354853679065601281848273721100803525082752733027258751379212217124487952578711033133791293778905903
1216476953628377088196350364071493632107992700239321497636297546469327503368318187732554219520137250097587144622081677501192731767603215377094266707869696105717523083616604165383762749628623360197404634153049523249351158538262380875397603322345304531649767016537477621599738215976982
132118465466505625602718697348682219520161989050358982246405939598513162682368014972737918597595136027450019517428924416037743776836464771072039541099542963093504031715256925084981248019381545898663044096088832085368872285440296106951229574284806785784299011077360956969580629767320626587225412347651912817993548893718485
1427116357971272007683728499221049901056023924536668403531776095033576210602917888026134233457915802419205270403747346353487872805200572511248449536094898638903111424409608699041899451880570880620209468757217411072034111520781646957608961421313365901956567040434290195136708951040918690797404576627201203047472791707488073519567781715457646927801441732859226832
Technical details
Node id
n-faaa8ff869da19e7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-60cc3cc9b4461a16

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:           264,            242;
  1:          4752,           8712,            3993;
  2:         76032,         209088,          191664,           58564;
  3:       1140480,        4181760,         5749920,         3513840,           805255;
  4:      16422912,       75271680,       137998080,       126498240,         57978360,         10629366;
  5:     229920768,     1264564224,      2897959680,      3541950720,       2435091120,        892866744,       136410197;
  6:    3153199104,    20233027584,     55640825856,     85006817280,      77922915840,      42857603712,     13095378912,      1714871048;
  7:   42568187904,   312166711296,   1001534865408,   1836147253248,    2103918727680,    1542873733632,    707150461248,    185206073184,     21221529219;
  8:  567575838720,  4682500669440,  17169169121280,  36722945064960,   50494049464320,   46286212008960,  28286018449920,  11112364391040,   2546583506280,   259374246010;
  9: 7492001071104, 68676676485120, 283291290501120, 692489821224960, 1110869088215040, 1221955997036544, 933438608847360, 488944033205760, 168074511414480, 34237400473320, 3138428376721;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 264 + 242*x.
%t Mathematica
P[0, x_] := 264 + 242*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (11*n + 22)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 264 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (11*n + 22)*x)*v[i - 1])/(n + 1)); 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.