Real-rooted atlas

A046739 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,n\right)\,P_{n}(x) = \left(-3\,n x^{2} - 5\,n^{2} x^{2} - 2\,n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-6\,x + 5\,x^{2} - 10\,n x + 3\,n x^{2} - 4\,n^{2} x - 6\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(-3\,x^{2} + 3\,x^{3} - 5\,n x^{2} + 5\,n x^{3} - 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-3}'(x) + \left(-6 + 10\,x + x^{2} - 4\,n + 8\,n x + 2\,n x^{2} - n^{2} x^{2} - 2\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x + 2\,x^{2} + x^{3} - 2\,n x - n x^{2} + 3\,n x^{3} - 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(5 + 6\,n + 2\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(x - x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 7 + 2\,x\)
\(P_{2}(x) = 21 + 42\,x + 3\,x^{2}\)
\(P_{3}(x) = 51 + 322\,x + 153\,x^{2} + 4\,x^{3}\)
\(P_{4}(x) = 113 + 1626\,x + 2439\,x^{2} + 452\,x^{3} + 5\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
1729
22142366
3513221534530
41131626243945254635
5239672222893134441195644499
6493248421614992153326210529587467236
710038573096032124280361600535257190702185339844
82025283098511407921976068274700851022815899084316200966080565
94071907810252680371716964523630265152575446785895875336312403663910881074205
1081652853450118423347120983325240101440654812172878211720819131579559212840525816501112599361126
1116355884582653419743379082257323894926425573033633926545289699571581645146416025922994422913017990512192426606294
12327372715861823426683114887433024434383520692595251879462211112719270595501363310801099672430497808894370149372399392844133126932352271
1365503828176661005855945328941473510828222196020751108213855373419058439426649147761847383125079995283735723536837770368813846614969059968515391453879449762215
14131037251334666425009891311657987550756218902568250651180432410739982866103210603433275546526403921770648616109974378051365013045594657645791700039565241633675329183451815981375692097480
1526210776016221017739383499392546155796521623614173327751172827357491174388984946232844562662619992448085001235022993715195471226248529035719511813210527763846738956768706618303532113547039316051618842413288271624
Technical details
Node id
n-912757bc57ce4c3d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ad948b5354047a3c

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    7,       2;
   2:   21,      42,         3;
   3:   51,     322,       153,          4;
   4:  113,    1626,      2439,        452,          5;
   5:  239,    6722,     22893,      13444,       1195,          6;
   6:  493,   24842,    161499,     215332,      62105,       2958,          7;
   7: 1003,   85730,    960321,    2428036,    1600535,     257190,       7021,         8;
   8: 2025,  283098,   5114079,   21976068,   27470085,   10228158,     990843,     16200,        9;
   9: 4071,  907810,  25268037,  171696452,  363026515,  257544678,   58958753,   3631240,    36639,    10;
  10: 8165, 2853450, 118423347, 1209833252, 4010144065, 4812172878, 2117208191, 315795592, 12840525, 81650, 11;
%F Formula
The row polynomials satisfy (2*n+1)*P(n,x) = -(2*n^3+5*n^2+3*n)*x^2*P(n-4,x) - ((4*n^2+10*n+6)*x+(4*n^3+6*n^2-3*n-5)*x^2)*P(n-3,x) - ((2*n^2+5*n+3)*x^2-(2*n^2+5*n+3)*x^3)*d/dx P(n-3,x) - (4*n+6-(8*n+10)*x+(2*n^3+n^2-2*n-1)*x^2)*P(n-2,x) - ((2*n+3)*x+(2*n^2+n-2)*x^2-(2*n^2+3*n+1)*x^3)*d/dx P(n-2,x) + (6*n+5+(4*n^2+2*n)*x)*P(n-1,x) + ((2*n+1)*x-(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 5, with P(1,x) = 7 + 2*x, P(2,x) = 21 + 42*x + 3*x^2, P(3,x) = 51 + 322*x + 153*x^2 + 4*x^3, P(4,x) = 113 + 1626*x + 2439*x^2 + 452*x^3 + 5*x^4.
%t Mathematica
P[1, x_] := 7 + 2*x;
P[2, x_] := 21 + 42*x + 3*x^2;
P[3, x_] := 51 + 322*x + 153*x^2 + 4*x^3;
P[4, x_] := 113 + 1626*x + 2439*x^2 + 452*x^3 + 5*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 5*n^2 + 3*n)*x^2*P[n - 4, x] - ((4*n^2 + 10*n + 6)*x + (4*n^3 + 6*n^2 - 3*n - 5)*x^2)*P[n - 3, x] - ((2*n^2 + 5*n + 3)*x^2 - (2*n^2 + 5*n + 3)*x^3)*D[P[n - 3, x], x] - (4*n + 6 - (8*n + 10)*x + (2*n^3 + n^2 - 2*n - 1)*x^2)*P[n - 2, x] - ((2*n + 3)*x + (2*n^2 + n - 2)*x^2 - (2*n^2 + 3*n + 1)*x^3)*D[P[n - 2, x], x] + (6*n + 5 + (4*n^2 + 2*n)*x)*P[n - 1, x] + ((2*n + 1)*x - (2*n + 1)*x^2)*D[P[n - 1, x], x])/(2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 7 + 2*x; v[2] = 21 + 42*x + 3*x^2; v[3] = 51 + 322*x + 153*x^2 + 4*x^3; v[4] = 113 + 1626*x + 2439*x^2 + 452*x^3 + 5*x^4; for(n = 5, N, my(i = n); v[i] = (-(2*n^3 + 5*n^2 + 3*n)*x^2*v[i - 4] - ((4*n^2 + 10*n + 6)*x + (4*n^3 + 6*n^2 - 3*n - 5)*x^2)*v[i - 3] - ((2*n^2 + 5*n + 3)*x^2 - (2*n^2 + 5*n + 3)*x^3)*deriv(v[i - 3], x) - (4*n + 6 - (8*n + 10)*x + (2*n^3 + n^2 - 2*n - 1)*x^2)*v[i - 2] - ((2*n + 3)*x + (2*n^2 + n - 2)*x^2 - (2*n^2 + 3*n + 1)*x^3)*deriv(v[i - 2], x) + (6*n + 5 + (4*n^2 + 2*n)*x)*v[i - 1] + ((2*n + 1)*x - (2*n + 1)*x^2)*deriv(v[i - 1], x))/(2*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.