Real-rooted atlas

A008278 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3117119
44946131109
5257322139231742
61539242714293784115815
7102991964514744543899175151193
875905170236155822741781926125741411498118
96094411573732170391499764633867165938670727115296321
1052844511545963719361158134921845683129146644922262917650529160987817
1149134923160769075228925579185388856935319533022212761701587500384709510431754940848
124870269291763691946281690816326030963791533391979597306260154161540255935602538749127686206919983845261
13512090544120346588710360508936293747506289325282270022115537901723618685680766948592105599861866769035197941191140329768789
14568780920672461361620314794252887555541694529624217797957322215685050828181181784021184655616376165540743591093529913797985866821712087182244308
15664920021819311429502967366176677347148425507433702714763122545842484308748981810005277690555306225312121314829431183220040171807124804104472782280142316411132725315072135
Technical details
Node id
n-f42aa320b67cb22d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-75d63132b26da3ac

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,        1;
   3:      11,        7,        1;
   4:      49,       46,       13,        1;
   5:     257,      322,      139,       23,       1;
   6:    1539,     2427,     1429,      378,      41,       1;
   7:   10299,    19645,    14744,     5438,     991,      75,      1;
   8:   75905,   170236,   155822,    74178,   19261,    2574,    141,     1;
   9:  609441,  1573732,  1703914,   997646,  338671,   65938,   6707,   271,   1;
  10: 5284451, 15459637, 19361158, 13492184, 5683129, 1466449, 222629, 17650, 529, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(n-1)*x)*P(n-1,x) - (4+4*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + (n - 1)*x)*P[n - 1, x] - (4 + 4*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 1 + (n - 1)*x)*v[i - 1] - (4 + 4*x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.