Real-rooted atlas

A201637 → reverse (shift +1) → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2123
318615
41225824105
5152328444120945
6111414524400370872010395
7124056103212058140339845040135135
8149419950195800644020785304341136403202027025
91100467260106250057655001244006411026296373392036288034459425
1012026218848532616044765000155357384238904904162186912443390403628800654729075
11140726950382524390431436972016483843044002695088464216395225074812165683564803991680013749310575
121816621706261148763762051482776155489607845604139878410118043302492199790224407883018247827719040479001600316234143225
1311635666996965072592761266981777613432342022468772004638418181886423042549865473424188307966182469792943680011533608576062270208007905853580625
1413273820507988218982980875016052228108467651241676348321493922829991006611257494373464592647283751398723968957805580812550904017920181099255680087178291200213458046676875
Technical details
Node id
n-db6c83a2e7fef454 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-36df5d4b0f5901bf

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2: 1,    2;
  3: 1,    8,     6;
  4: 1,   22,    58,      24;
  5: 1,   52,   328,     444,     120;
  6: 1,  114,  1452,    4400,    3708,      720;
  7: 1,  240,  5610,   32120,   58140,    33984,     5040;
  8: 1,  494, 19950,  195800,  644020,   785304,   341136,   40320;
  9: 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n-2)*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[(1 + (2*n - 2)*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] = (1 + (2*n - 2)*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. [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