Real-rooted atlas

A130850 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
265112
3262910166
4150209101191480
51082180911263143614368
693661826913917515893269147712
794586210853190002879192180627151341608016
8109167027377692848477158099749966988331786326318855040
914174522394976574659789430170960116006862668982349642227845201145083648
102044951266268138138268236696118582242773123887875332213725218136746466254103312641216512
113245265146108516141571582616749013177070829688045645223277573545082763086893349053187541935772058152891055616
12561831351902035224090893251763455733008740857191778815413456989415210618378890466317135825434118306420653125568361410711155444326400
131053716696762411069078422571405513062147268746927766480014091496021492228596796617853250201390500955081933875940216738949758024870216763108204127344999497728
14212826859408868895709813543716691862736503718539167998244213526624224799667999882425187240141576501995958213118284102057559661411610147734781720986513124163624963272163971696933753434112
Technical details
Node id
n-6064233ba865bfe5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-179ce8abeeb2cd96

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,        1;
  2:        6,        5,        1;
  3:       26,       29,       10,        1;
  4:      150,      209,      101,       19,        1;
  5:     1082,     1809,     1126,      314,       36,       1;
  6:     9366,    18269,    13917,     5158,      932,      69,      1;
  7:    94586,   210853,   190002,    87919,    21806,    2715,    134,     1;
  8:  1091670,  2737769,  2848477,  1580997,   499669,   88331,   7863,   263,   1;
  9: 14174522, 39497657, 46597894, 30170960, 11600686, 2668982, 349642, 22784, 520, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+n*x)*P(n-1,x) - (2+3*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 + n*x)*P[n - 1, x] - (2 + 3*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 + n*x)*v[i - 1] - (2 + 3*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.