Real-rooted atlas

A114687 → basis A013610 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(6 + 3\,n\right)\,P_{n}(x) = \left(6 + 75\,n + 54\,n x\right)\,P_{n-1}(x) + \left(-56 - 90\,x - 36\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19615
28911436239
3945176410802164005
41057725788231849072129669917
51231293696904366802527207128077761261275
6147684152621747689060587844024688805365444665623358595
7181304017472925613003502412348504068856480224259843919104279936441861225
822673948910609568882141407296242932435216889523847337433601933424642799360016796168504139449
9287866685715072221070346105807204565665440038030005440206713140487297371648160235366419651507210077696166025760615
103700632677721433873041055171599930082967123880080543302560052592742720023304951936068839061760128742566401360488960604661763280276540983
114807509909933051655523532870213741012014692931253000162902435256001242355314816066299090457602465684686080622726030080100877736960931179110436279705665470143938445
12630161163147343500908557908136117916420112254955258938160317858594288400277327941264000173195742551040777540099916802479513804416054440721561607742089175046312668774421767823361318090706230677
13832406695829376208407611408342114879392232568435171447398001660262619401876325920721398376928423699857529062422301459231800328603782924239362391681443374084625964794880058388558192644244725555201306069401626736885607750515
141106980509493641887074228201455032678159666426820732738001298612401115914020409188001219249573648666569838353309226041659428581644285568269279147248151049074729198126592222792517185638438372790655180843381095174144283417060147278364164096545918747361417291

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(2 + n\right)\,P_{n}(x) = \left(49 + 84\,x + 36\,x^{2} - 49\,n - 84\,n x - 36\,n x^{2}\right)\,P_{n-2}(x) + \left(9 + 6\,x + 18\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 6\,x\)
Technical details
Node id
n-ce8106c936a63397 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-e0757929e12c07e5)
Fingerprint
f6-807d8dba3ebdc567

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          9,           6;
  2:         89,         114,          36;
  3:        945,        1764,        1080,         216;
  4:      10577,       25788,       23184,        9072,        1296;
  5:     123129,      369690,      436680,      252720,       71280,        7776;
  6:    1476841,     5262174,     7689060,     5878440,     2468880,      536544,      46656;
  7:   18130401,    74729256,   130035024,   123485040,    68856480,    22425984,    3919104,     279936;
  8:  226739489,  1060956888,  2141407296,  2429324352,  1688952384,   733743360,  193342464,   27993600,   1679616;
  9: 2878666857, 15072221070, 34610580720, 45656654400, 38030005440, 20671314048, 7297371648, 1602353664, 196515072, 10077696;
%F Formula
The row polynomials satisfy (3*n+6)*P(n,x) = (75*n+6+54*n*x)*P(n-1,x) - (56+90*x+36*x^2)*d/dx 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[((75*n + 6 + 54*n*x)*P[n - 1, x] - (56 + 90*x + 36*x^2)*D[P[n - 1, x], x])/(3*n + 6)]];
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] = ((75*n + 6 + 54*n*x)*v[i - 1] - (56 + 90*x + 36*x^2)*deriv(v[i - 1], x))/(3*n + 6)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023