Real-rooted atlas

A074246 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 2\,x - x^{2} - 2\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(3 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22618347
3154142424342
4104411604658052754
580281020849951180135624552
66926497720552721610025552107241128
7663696101800864402821859643120495630882592720
8699984011507472796729230446647044451013048862432930334320
980627040140581872104901420441143601155472519596782156701488058510383970240
1010074412801848236544146968984866924088019379965037381806487433142504023760770115231113920
1113575738240260419576322187832284010659717904335594655071852305810697841011096184787050364109901276349105280
1219628735616039173888332834535945620817840714777660307899080140864586722335974641277607616235019071384240539111248131188825724800
13303148863360062686579622405768000168112313658649158411276274209202836663589765147382139968337995446648371734687683023313297753215471419675048780800
1449811492505600106361504663040101690989633152578760507561602196277184920058918146487201154525227856168168017160183254192551488737250888657773783780108745189015344937224217600
Technical details
Node id
n-36740f0bed8145fa · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6443481be5042dac

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:        5,         2;
  2:       26,        18,         3;
  3:      154,       142,        42,        4;
  4:     1044,      1160,       465,       80,        5;
  5:     8028,     10208,      4995,     1180,      135,       6;
  6:    69264,     97720,     55272,    16100,     2555,     210,      7;
  7:   663696,   1018008,    644028,   218596,    43120,    4956,    308,     8;
  8:  6999840,  11507472,   7967292,  3044664,   704445,  101304,   8862,   432,   9;
  9: 80627040, 140581872, 104901420, 44114360, 11554725, 1959678, 215670, 14880, 585, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 2*x)*P[n - 1, x]];
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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 2*x)*v[i - 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

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. [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.