Real-rooted atlas

A144442 → D

nonnegative coefficients

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

Recurrence

\(\left(-4 + 5\,n\right)\,P_{n}(x) = \left(-25\,x^{2} - 125\,n x^{2}\right)\,P_{n-4}(x) + \left(-10\,x + 15\,x^{2} - 50\,n x + 650\,n x^{2} - 250\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-25\,x^{2} + 25\,x^{3} - 125\,n x^{2} + 125\,n x^{3}\right)\,P_{n-3}'(x) + \left(-6 - 12\,x + 224\,x^{2} - 30\,n + 180\,n x - 640\,n x^{2} - 50\,n^{2} x + 550\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - 75\,x^{2} + 80\,x^{3} - 25\,n x + 225\,n x^{2} - 200\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-23 + 72\,x + 35\,n - 130\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 20\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 17 + 2\,x\)
\(P_{3}(x) = 118 + 236\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
217219
31182363357
47295362218748282
5440083490125235176005230730
62643111104804047735222096013215567537767
715862213633692100370055133826740340842309517327283025078
89517731600694862145963468564770798035766057804802084586662411812018169364
95710684183087822241880899277193664401504242080501880837617985546408073777456854729569677949379
10342641552061745426077016233356558123360248201262348427263587185040372301797045444985824698170403083773951029824962026095
11205584986230001405960136043023606951593060545275205546490145596856655788174716222787855954231603627813962852010350063268202055849860111709469193138839
121233509977255132777568223369356568931040942634230196802162597393244533541341627588876374302763635054234698188526846039360701080697067930127566388784101356860974712106476852327335286
1374010599282820481449978639359986364133331004034325947622607731019855141343102219160423224164176258902049376152487212369631768226148962259077233382150851311999545471111015512647974882388812719136137161796656854567728
144440635963931116822390863265364317558912967237767794887070950425899438143960195010106960346650839169518186650635045098236299142613795534452226024466189886591283510185944194872176773760239669164382680879186700934345179257728267530714517372289135723650363
Technical details
Node id
n-f7d4ac866eaf7630 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e45b34e29a2fbcd1

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:       1;
  2:      17,          2;
  3:     118,        236,           3;
  4:     729,       5362,        2187,            4;
  5:    4400,      83490,      125235,        17600,            5;
  6:   26431,    1110480,     4047735,      2220960,       132155,           6;
  7:  158622,   13633692,   100370055,    133826740,     34084230,      951732,          7;
  8:  951773,  160069486,  2145963468,   5647707980,   3576605780,   480208458,    6662411,        8;
  9: 5710684, 1830878222, 41880899277, 193664401504, 242080501880, 83761798554, 6408073777, 45685472, 9;
%F Formula
The row polynomials satisfy (5*n-4)*P(n,x) = -(125*n+25)*x^2*P(n-4,x) - ((50*n+10)*x+(250*n^2-650*n-15)*x^2)*P(n-3,x) - ((125*n+25)*x^2-(125*n+25)*x^3)*d/dx P(n-3,x) - (30*n+6+(50*n^2-180*n+12)*x+(125*n^3-550*n^2+640*n-224)*x^2)*P(n-2,x) - ((25*n+5)*x+(125*n^2-225*n+75)*x^2-(125*n^2-200*n+80)*x^3)*d/dx P(n-2,x) + (35*n-23+(50*n^2-130*n+72)*x)*P(n-1,x) + ((25*n-20)*x-(25*n-20)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 17 + 2*x, P(3,x) = 118 + 236*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 17 + 2*x;
P[3, x_] := 118 + 236*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(125*n + 25)*x^2*P[n - 4, x] - ((50*n + 10)*x + (250*n^2 - 650*n - 15)*x^2)*P[n - 3, x] - ((125*n + 25)*x^2 - (125*n + 25)*x^3)*D[P[n - 3, x], x] - (30*n + 6 + (50*n^2 - 180*n + 12)*x + (125*n^3 - 550*n^2 + 640*n - 224)*x^2)*P[n - 2, x] - ((25*n + 5)*x + (125*n^2 - 225*n + 75)*x^2 - (125*n^2 - 200*n + 80)*x^3)*D[P[n - 2, x], x] + (35*n - 23 + (50*n^2 - 130*n + 72)*x)*P[n - 1, x] + ((25*n - 20)*x - (25*n - 20)*x^2)*D[P[n - 1, x], x])/(5*n - 4)]];
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] = 1; v[3] = 17 + 2*x; v[4] = 118 + 236*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(125*n + 25)*x^2*v[i - 4] - ((50*n + 10)*x + (250*n^2 - 650*n - 15)*x^2)*v[i - 3] - ((125*n + 25)*x^2 - (125*n + 25)*x^3)*deriv(v[i - 3], x) - (30*n + 6 + (50*n^2 - 180*n + 12)*x + (125*n^3 - 550*n^2 + 640*n - 224)*x^2)*v[i - 2] - ((25*n + 5)*x + (125*n^2 - 225*n + 75)*x^2 - (125*n^2 - 200*n + 80)*x^3)*deriv(v[i - 2], x) + (35*n - 23 + (50*n^2 - 130*n + 72)*x)*v[i - 1] + ((25*n - 20)*x - (25*n - 20)*x^2)*deriv(v[i - 1], x))/(5*n - 4)); 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.