Real-rooted atlas

A034870 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 3\,x + 6\,n\right)\,P_{n}(x) = \left(8 + 36\,x + 54\,x^{2} + 27\,x^{3} + 24\,n + 72\,n x + 54\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11630955
2112528864540812125
36404896151202376019440729072971875
43328353281612804112646350405987523265928748065612265625
516384222720134784047692801088640016656192171460801154736047239209841505904968359375
677824129024097320964409856013343616028329523243110144047113228836374184019053144062355744106288205314412001953125
736044870533126373785635223552013284311043611672064728561433611048514048126097171201071825955266201014882837186352773778096111602610478296957373046875
816384003696230438928384025391923201147281408038041436160956470394881857431347202810165529603314554214403026332108802104676421121083289334403928411872091833004801147912560430467211617431640625
97340032187564032225607680016965697536893406412803496467824641053204332544249443131392047037847633927104674903040859910652057682984384604166314029263360371556337420816392191356805151831569281053783730081162261467038742048945013427734375
103250585692798976012551454720106911498240642981888000290113427865610184833105920284712380006406432069316608011848548741120017874382100889622089080153088022279503257856018192730470912011864824220160060510603522816233588726834406477670576080117775828656011622614670034867844011239776611328125
111426063364498391040675796746246430969036804349079060480222269004840968911728967680028721229358694475622385811456016442139945369602971616024272896447875142414336056304303617802245887589440803840509000959044864036021606332405762055580796142720925011358264224317405858227920787548370039201288715514609611506388523303138105960933855438232421875
126207569922144128204835423413862437246677811202797798529433615974527087411272020749241548826291248001187847905976698470400198176482574991364174355696792371274258682953269248111840821277179904142631006891950080153692187018264576139283544485302272105363198522286848657924045071293443337295880796416013437007098996096414966395704291293155721484636813857875923334411297181459240282429536481917911529541015625
13268435456010076395929618157142016002090524999680017268914257920010894653259776005455707132395520222502341574656007523596126126080021361639000965120051388848540057600010539385664215449601849998547442073600278470061074759680035948835410522112003973292334847180800374624705857019904029956203501434496002014296442337836800112539639288510720051376791849102720018715831316458848052544446209309600107699796578088001468633589701200110147519227590254186582832924735927581787109375
14115427246084678829998089132174213120114266329841664102950995230720071119660489113603916589753696256017649338280837120066286841658605568021026471278804992005687838671351316480132131447056878796802648861556564787200045967935935462768640691614841582190592009022429978822213632010191494838883018752099390739140809625600832989051846785433605959707854495915520036066507878070109440182314215647387366407566400254140582400251665051931197632064766741305822920012213156931955179215373204529733792106758364789818022876792454961663101673126220703125
Technical details
Node id
n-ef2d25c676dbb756 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7939c1a278d41502

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:       1;
  1:      16,        30,          9;
  2:     112,       528,        864,         540,          81;
  3:     640,      4896,      15120,       23760,       19440,         7290,           729;
  4:    3328,     35328,     161280,      411264,      635040,       598752,        326592,         87480,          6561;
  5:   16384,    222720,    1347840,     4769280,    10886400,     16656192,      17146080,      11547360,       4723920,        984150,         59049;
  6:   77824,   1290240,    9732096,    44098560,   133436160,    283295232,     431101440,     471132288,     363741840,     190531440,      62355744,      10628820,        531441;
  7:  360448,   7053312,   63737856,   352235520,  1328431104,   3611672064,    7285614336,   11048514048,   12609717120,   10718259552,    6620101488,    2837186352,     773778096,     111602610,       4782969;
  8: 1638400,  36962304,  389283840,  2539192320, 11472814080,  38041436160,   95647039488,  185743134720,  281016552960,  331455421440,  302633210880,  210467642112,  108328933440,   39284118720,    9183300480,   1147912560,     43046721;
  9: 7340032, 187564032, 2256076800, 16965697536, 89340641280, 349646782464, 1053204332544, 2494431313920, 4703784763392, 7104674903040, 8599106520576, 8298438460416, 6314029263360, 3715563374208, 1639219135680, 515183156928, 105378373008, 11622614670, 387420489;
%F Formula
The row polynomials satisfy (6*n-4+3*x)*P(n,x) = (24*n+8+(72*n+36)*x+(54*n+54)*x^2+27*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 8 + (72*n + 36)*x + (54*n + 54)*x^2 + 27*x^3)*P[n - 1, x])/(6*n - 4 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 8 + (72*n + 36)*x + (54*n + 54)*x^2 + 27*x^3)*v[i - 1])/(6*n - 4 + 3*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.