Real-rooted atlas

A191935 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
248113
385220181
416320292401669
5321936382410927016955
6641164847824256643192112188505
712869952585536561104121424789616811346209
8256419840709638411807616420382445305617304240123998521
9512251929685576448243248704137922336232321761422080345843301493956467
101024151162881029436672495055052843809187841113673728102868656391212864284440111596542533
112048906987521236835635210004044723213637811404851155215360687431846438454187296976881126845721307301505073
1240965441945601485109749762013177300992419138386867222848971591684361170493443456133324812573337922209292818818872819110471500693
13819232651714561782675894272404120569948161277546933611521001570645537282670745192243229245330432001477213745283687555872469337443017569101299893197116059
14163841959103692821395375902720810023815790592387305285782937643343514046177281595774372209920237273831032832162194567507205533525204489882810080940076804673761120110888674034993905
Technical details
Node id
n-475d7faad9200466 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-57cb71c89ecc0cb7

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:   4,       8,        1;
  3:   8,      52,       20,         1;
  4:  16,     320,      292,        40,         1;
  5:  32,    1936,     3824,      1092,        70,        1;
  6:  64,   11648,    47824,     25664,      3192,      112,       1;
  7: 128,   69952,   585536,    561104,    121424,     7896,     168,     1;
  8: 256,  419840,  7096384,  11807616,   4203824,   453056,   17304,   240,   1;
  9: 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + 4*x*d/dx P(n-1,x) + x^2*d^2/dx^2 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 + x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
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 + x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.