Real-rooted atlas

A081580 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(18\,x^{2} + 6\,x^{3} + 8\,n - 10\,n x - 30\,n x^{2} - 9\,n x^{3} + 14\,n^{2} x + 12\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(74\,x^{2} + 66\,x^{3} + 12\,x^{4} - 8\,n x - 52\,n x^{2} - 36\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}'(x) + \left(8\,x^{2} + 50\,x^{3} + 24\,x^{4} + 3\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21636961
36425616828516
4256147219848081014621
5102476801632011840342034240626
640963788811136012704063440142681189359281
71638418022467993610956808262403115525723240883171336

All 15 rows

Technical details
Node id
n-6c47aa5f31984af6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-679f9f4065461fd6

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,      36,        9;
  3:     64,     256,      168,       28;
  4:    256,    1472,     1984,      808,      101;
  5:   1024,    7680,    16320,    11840,     3420,      342;
  6:   4096,   37888,   111360,   127040,    63440,    14268,     1189;
  7:  16384,  180224,   679936,  1095680,   826240,   311552,    57232,    4088;
  8:  65536,  835584,  3858432,  8159232,  8593920,  4802688,  1454592,  225936,  14121;
  9: 262144, 3801088, 20791296, 54853632, 76007424, 58401792, 25785984, 6515328, 876276, 48682;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (8*n+(14*n^2-10*n)*x+(12*n^2-30*n+18)*x^2+(3*n^2-9*n+6)*x^3)*P(n-1,x) - (8*n*x+(52*n-74)*x^2+(36*n-66)*x^3+(6*n-12)*x^4)*d/dx P(n-1,x) + (8*x^2+50*x^3+24*x^4+3*x^5)*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[Together[((8*n + (14*n^2 - 10*n)*x + (12*n^2 - 30*n + 18)*x^2 + (3*n^2 - 9*n + 6)*x^3)*P[n - 1, x] - (8*n*x + (52*n - 74)*x^2 + (36*n - 66)*x^3 + (6*n - 12)*x^4)*D[P[n - 1, x], x] + (8*x^2 + 50*x^3 + 24*x^4 + 3*x^5)*D[P[n - 1, x], {x, 2}])/(2*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] = ((8*n + (14*n^2 - 10*n)*x + (12*n^2 - 30*n + 18)*x^2 + (3*n^2 - 9*n + 6)*x^3)*v[i - 1] - (8*n*x + (52*n - 74)*x^2 + (36*n - 66)*x^3 + (6*n - 12)*x^4)*deriv(v[i - 1], x) + (8*x^2 + 50*x^3 + 24*x^4 + 3*x^5)*deriv(deriv(v[i - 1], x), x))/(2*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