Real-rooted atlas

A059364 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
21311226
370104476227
4446984767242242463
532769764111676017147412031818
6272521033401583771242475166410404720476004
725329611729682274344237751214317494874688362850408086005
826022241428688833692808443772883546462917420244503918075470440320153678285
9292881601864429925197240208300152608331390035414010092253015445678895675584163628803230022240
10358457760260034811283853123001579184874019173609613156112935398573913354310218383669395709683204640362880074377757790
11474057792038651163264141777516168308303452168440580527534433069794152297427799675141753029554454402880449148066776998713440399168001861930845495
12673755321606104909733122513259191016620760596298410227515917766118142013937049779983890683581942992759024534512751407068296891281294756197121298248128047900160050343080856075
13102420872064010217686192128466927373299681292794007918242416336827905043213981513444083122615581238072236664702855371175756479529144449271839511611650099613448195898580121618170529984062270208001462039279662150
141658362688640018072264098304090840325697812827891339119238245837833611357752879973478936365698341113504606898258237147152271521993434139892824602376287378808452605793394722029853058776163156193712678427244578009608717829120045387551763678600
Technical details
Node id
n-313842148b1bb9e8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7f00433e786f3a7b

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:        3,         1;
  2:       13,        11,         2;
  3:       70,       104,        47,         6;
  4:      446,       984,       767,       242,        24;
  5:     3276,      9764,     11167,      6017,      1474,       120;
  6:    27252,    103340,    158377,    124247,     51664,     10404,       720;
  7:   253296,   1172968,   2274344,   2377512,   1431749,    487468,     83628,     5040;
  8:  2602224,  14286888,  33692808,  44377288,  35464629,  17420244,   5039180,   754704,   40320;
  9: 29288160, 186442992, 519724020, 830015260, 833139003, 541401009, 225301544, 56788956, 7558416, 362880;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(2*n^3-2*n^2)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n^2 - 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.