Real-rooted atlas

A099040 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4\,x - 3\,x^{2} + 4\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} - 3\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21620642
3641288016288
4256704688288441980
5102435844800310497612013608
6409617408296962617612640318432893528
716384819201699841904641249284812810112896642816
865536376832921600125542410447365452801747203148824484418064
926214417039364800512770867277916165151232223180861209696576668830365280
101048576760217624248320448757765344870442877952235002888701440208588829267218272208700064
11419430433554432119537664250609664343982080324993024215945216101036032326584326952960878336499201434392064
1216777216146800640577765376135371161621051473922291482624179217612810157916164145121281188751362276172826147841363849858552768
13671088646375342082747269120711563673612369526784152533729281373960601691612856324525666304163723059242192588873401856773145637260867757668992
142684354562751463424128849018883656174796870279757824968805580809884139520075876024320440727552001928988672062666895361466256384233712128227284481017984465697330560
Technical details
Node id
n-5dc7ef46a24aaa48 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5143ff9180de8713

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:      4,       2;
  2:     16,      20,       6;
  3:     64,     128,      80,      16;
  4:    256,     704,     688,     288,      44;
  5:   1024,    3584,    4800,    3104,     976,     120;
  6:   4096,   17408,   29696,   26176,   12640,    3184,     328;
  7:  16384,   81920,  169984,  190464,  124928,   48128,   10112,    896;
  8:  65536,  376832,  921600, 1255424, 1044736,  545280,  174720,  31488,  2448;
  9: 262144, 1703936, 4800512, 7708672, 7791616, 5151232, 2231808, 612096, 96576, 6688;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n+(6*n-4)*x+(3*n-3)*x^2)*P(n-1,x) - (4*x+8*x^2+3*x^3)*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[Together[((4*n + (6*n - 4)*x + (3*n - 3)*x^2)*P[n - 1, x] - (4*x + 8*x^2 + 3*x^3)*D[P[n - 1, x], x])/(n)]];
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] = ((4*n + (6*n - 4)*x + (3*n - 3)*x^2)*v[i - 1] - (4*x + 8*x^2 + 3*x^3)*deriv(v[i - 1], x))/(n)); 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
  2. [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