Real-rooted atlas

A112486 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
235210
3153526682
410531534015424938
5945346549003304104412013778
610395450457875070532337408028720247210
71351356756751406790157157010089803678846926450405240338
820270251148647527747720371471102995762014777620430221666369640320128149802
9344594252182430255999994009362753409095086005667805002246558165396133669998403628803551246162
10654729075458310352514128364250251803952402862303444021577556000108554526083568061640724534272806270403628800109979486890
111374931057510541138107536010099125072168603507093983869980083183248148050807268512021278465208059371808496103864700161007441280399168003764281873042
123162341432252635284526875988117119990021995841517650322906559976003282991996284023610220313680120417900803204295131560720103598770766415850731686413575738240479001600141104799067178
137905853580625711526822256252905137662427007111754509860001162012289337900133470217520640011051925852167206653464074056802897694579780008959303985486418953413075584256767008832019628735616062270208005749087305575378
14213458046676875206342778454312591128138939337502433105497973150043794493478435700560897305093125005257249214409192036515972437664720188125478496956007118294418969552193488273967248036329074369804844027031755520303148863360087178291200252969604725106090
Technical details
Node id
n-95a09e5daac73697 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9d05ffe77ba48b7a

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:        1,         1;
  2:        3,         5,         2;
  3:       15,        35,        26,         6;
  4:      105,       315,       340,       154,        24;
  5:      945,      3465,      4900,      3304,      1044,       120;
  6:    10395,     45045,     78750,     70532,     33740,      8028,       720;
  7:   135135,    675675,   1406790,   1571570,   1008980,    367884,     69264,     5040;
  8:  2027025,  11486475,  27747720,  37147110,  29957620,  14777620,   4302216,   663696,   40320;
  9: 34459425, 218243025, 599999400, 936275340, 909508600, 566780500, 224655816, 53961336, 6999840, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (x+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[(2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
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] = (2*n - 1 + (2*n - 1)*x)*v[i - 1] - (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 - 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. [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.