Real-rooted atlas

A059364 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 2\,n x - n x^{2} + 4\,n^{2} + 6\,n^{2} x + 2\,n^{2} x^{2} - 4\,n^{3} - 4\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 2\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
22615243
3227216656514
4246333401637338247802
5318185664539262131132074120143032
64760041055831955942449343114484147247203067048
7808600521555576242323141482814852999091095076118908504075220976
815367828547960467264638033048989440422739058965762028114510681077264403202075278960
93230022240115685630731821940447216523978918948209266835547725258649418041303199641082433636288063585282880
10743777577903009929785835432594741825749401670583942380412381824604738355747123665412084663644160609485611949264036288002141554009280
1118619308454958407906762464171278955931232075076115616216585081471065916188010555635586360649699676238946421793012658282133062517614377982403991680078623825499520
125034308085607525108919963478057026553133880777909093827363071219166355602945816107724114421221670694376571093571392358170205365675322821002708648303897019392187305004804790016003124616415034240
13146203927966215079855523958488852001756676173557430467978603148783313846348223702002307508349522489512437586961724734496508294931414514634555525051583131654194332764697242116506446251570216962626565702406227020800133624012702745600
1445387551763678600269529706656681615739465967772261306124151302633036366514239201073601517721178996486963874713725844022630495578336974106659515983118282707016189928311431796799627606027647194926752825969157870704749131130384643944953877760871782912006117989424431180800
Technical details
Node id
n-f91294e5f7553d54 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-74619c273d5282fa

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:          4,           1;
  2:         26,          15,           2;
  3:        227,         216,          65,           6;
  4:       2463,        3340,        1637,         338,         24;
  5:      31818,       56645,       39262,       13113,       2074,        120;
  6:     476004,     1055831,      955942,      449343,     114484,      14724,       720;
  7:    8086005,    21555576,    24232314,    14828148,    5299909,    1095076,    118908,      5040;
  8:  153678285,   479604672,   646380330,   489894404,  227390589,   65762028,  11451068,   1077264,    40320;
  9: 3230022240, 11568563073, 18219404472, 16523978918, 9482092668, 3554772525, 864941804, 130319964, 10824336, 362880;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(4*n^3-6*n^2+2*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (4*n^3 - 6*n^2 + 2*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (2*n^2 - 2*n)*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] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (4*n^3 - 6*n^2 + 2*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (2*n^2 - 2*n)*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

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.