Real-rooted atlas

A145596 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(4\,n + n^{2}\right)\,P_{n}(x) = \left(4 + 8\,x + 5\,n + 10\,n x + n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2} + 2\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
23141431
34327248156
4560225330165785
56100550132014305723978
671541155400470076006200220335
782242184101922548034944247527072104856
893123822229327644014851216707610077625194545073
910420630047040199920514080813960775200406980904402854350
10115509900897604712401534896319770042636003517470163438032687615046383
11127041496016156810232644093056107442721875984021573816156900486537520118864079787700
12138842187927713420785059976824320089777011490210517235310623470069052555260757904345965425360481
131410923112245645639939902259457286612526230966736433062630568668100511801290300638520103791870159679802278586898
1415133043225726180732231548118070216531315687401000156383727525590064502985507525242181030012973983754125061505892945012259138975
Technical details
Node id
n-8b0588909446c2ef · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-db69b92786c145d5

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:  2,   4;
  2:  3,  14,   14;
  3:  4,  32,   72,    48;
  4:  5,  60,  225,   330,    165;
  5:  6, 100,  550,  1320,   1430,    572;
  6:  7, 154, 1155,  4004,   7007,   6006,   2002;
  7:  8, 224, 2184, 10192,  25480,  34944,  24752,   7072;
  8:  9, 312, 3822, 22932,  76440, 148512, 167076, 100776,  25194;
  9: 10, 420, 6300, 47040, 199920, 514080, 813960, 775200, 406980, 90440;
%F Formula
The row polynomials satisfy (n^2+4*n)*P(n,x) = (n^2+5*n+4+(2*n^2+10*n+8)*x)*P(n-1,x) + ((2*n+2)*x+(2*n+2)*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[((n^2 + 5*n + 4 + (2*n^2 + 10*n + 8)*x)*P[n - 1, x] + ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
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] = ((n^2 + 5*n + 4 + (2*n^2 + 10*n + 8)*x)*v[i - 1] + ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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

Candidate equalities ?

Same fingerprint ?

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.