Real-rooted atlas

A121574 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-44 - 32\,x + 64\,x^{2} + 56\,n - 16\,n x - 64\,n x^{2} - 12\,n^{2} + 48\,n^{2} x\right)\,P_{n}(x) = \left(-108 - 608\,x - 1088\,x^{2} - 512\,x^{3} + 316\,n + 1216\,n x + 832\,n x^{2} - 96\,n^{2} - 192\,n^{2} x\right)\,P_{n-1}(x) + \left(-26 - 236\,x - 720\,x^{2} - 832\,x^{3} - 256\,x^{4} + 39\,n + 276\,n x + 528\,n x^{2} + 192\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2283260
3140352192684
46202432288010246956
52568137282553619456512066408
6102006912017644821094411776024576609048
7393643232321051968172953615104006635521146885432740

All 15 rows

Technical details
Node id
n-747fc304024cb7fc · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-354136365d8a474b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      4;
  2:     28,      32;
  3:    140,     352,      192;
  4:    620,    2432,     2880,     1024;
  5:   2568,   13728,    25536,    19456,      5120;
  6:  10200,   69120,   176448,   210944,    117760,     24576;
  7:  39364,  323232,  1051968,  1729536,   1510400,    663552,   114688;
  8: 148756, 1435328,  5686464, 11951104,  14351360,   9830400,  3555328,   524288;
  9: 553220, 6134176, 28662720, 73521152, 112967680, 106291200, 59752448, 18350080, 2359296;
%F Formula
The row polynomials satisfy -(12*n^2-56*n+44-(48*n^2-16*n-32)*x+(64*n-64)*x^2)*P(n,x) = -(96*n^2-316*n+108+(192*n^2-1216*n+608)*x-(832*n-1088)*x^2+512*x^3)*P(n-1,x) + (39*n-26+(276*n-236)*x+(528*n-720)*x^2+(192*n-832)*x^3-256*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(96*n^2 - 316*n + 108 + (192*n^2 - 1216*n + 608)*x - (832*n - 1088)*x^2 + 512*x^3)*P[n - 1, x] + (39*n - 26 + (276*n - 236)*x + (528*n - 720)*x^2 + (192*n - 832)*x^3 - 256*x^4)*D[P[n - 1, x], x])/(-12*n^2 + 56*n - 44 + (48*n^2 - 16*n - 32)*x - (64*n - 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(96*n^2 - 316*n + 108 + (192*n^2 - 1216*n + 608)*x - (832*n - 1088)*x^2 + 512*x^3)*v[i - 1] + (39*n - 26 + (276*n - 236)*x + (528*n - 720)*x^2 + (192*n - 832)*x^3 - 256*x^4)*deriv(v[i - 1], x))/(-12*n^2 + 56*n - 44 + (48*n^2 - 16*n - 32)*x - (64*n - 64)*x^2)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.