Real-rooted atlas

A119308 → D

nonnegative coefficients

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

Recurrence

\(\left(12 + 34\,n + 22\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(-1304 + 2808\,x - 1704\,x^{2} + 200\,x^{3} - 1226\,n + 2226\,n x - 774\,n x^{2} - 226\,n x^{3} + 148\,n^{2} - 396\,n^{2} x + 348\,n^{2} x^{2} - 100\,n^{2} x^{3} + 60\,n^{3} - 84\,n^{3} x - 12\,n^{3} x^{2} + 36\,n^{3} x^{3}\right)\,P_{n-5}(x) + \left(420 - 1312\,x + 1416\,x^{2} - 576\,x^{3} + 52\,x^{4} + 375\,n - 1016\,n x + 798\,n x^{2} - 48\,n x^{3} - 109\,n x^{4} - 56\,n^{2} + 208\,n^{2} x - 288\,n^{2} x^{2} + 176\,n^{2} x^{3} - 40\,n^{2} x^{4} - 16\,n^{3} + 32\,n^{3} x - 32\,n^{3} x^{3} + 16\,n^{3} x^{4}\right)\,P_{n-5}'(x) + \left(-1516 + 114\,x + 600\,x^{2} - 118\,x^{3} - 704\,n - 704\,n x + 168\,n x^{2} + 168\,n x^{3} + 514\,n^{2} + 4\,n^{2} x - 294\,n^{2} x^{2} + 32\,n^{2} x^{3} - 52\,n^{3} + 208\,n^{3} x + 12\,n^{3} x^{2} - 40\,n^{3} x^{3}\right)\,P_{n-4}(x) + \left(-582 + 391\,x + 9\,x^{2} + 217\,x^{3} - 35\,x^{4} - 504\,n + 164\,n x + 176\,n x^{2} + 100\,n x^{3} + 64\,n x^{4} + 176\,n^{2} - 160\,n^{2} x + 56\,n^{2} x^{2} - 80\,n^{2} x^{3} + 8\,n^{2} x^{4} + 64\,n^{3} - 32\,n^{3} x - 16\,n^{3} x^{2} - 16\,n^{3} x^{4}\right)\,P_{n-4}'(x) + \left(336 - 330\,x + 60\,x^{2} + 6\,x^{3} - 338\,n + 430\,n x - 102\,n x^{2} - 14\,n x^{3} - 712\,n^{2} + 268\,n^{2} x - 24\,n^{2} x^{2} + 4\,n^{2} x^{3} - 72\,n^{3} - 152\,n^{3} x + 12\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(6 - 150\,x - 123\,x^{2} + 24\,x^{3} + 3\,x^{4} - 18\,n + 24\,n x + 71\,n x^{2} - 58\,n x^{3} - 3\,n x^{4} - 192\,n^{2} + 80\,n^{2} x + 272\,n^{2} x^{2} + 32\,n^{2} x^{3} - 96\,n^{3} - 32\,n^{3} x + 32\,n^{3} x^{2} + 32\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(8 + 24\,x - 22\,n + 100\,n x - 12\,n x^{2} + 36\,n^{2} + 76\,n^{2} x - 30\,n^{2} x^{2} + 56\,n^{3} + 16\,n^{3} x - 12\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(54 - 161\,x + 24\,x^{2} + 3\,x^{3} + 48\,n - 316\,n x + 14\,n x^{2} + 6\,n x^{3} + 80\,n^{2} - 128\,n^{2} x - 40\,n^{2} x^{2} + 64\,n^{3} + 32\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(-8 + 24\,x + 20\,n + 60\,n x + 36\,n^{2} + 48\,n^{2} x + 12\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(102 - 16\,x - 6\,x^{2} + 99\,n - 8\,n x - 3\,n x^{2} - 8\,n^{2} - 16\,n^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 2\,x\)
\(P_{3}(x) = 14 + 18\,x + 3\,x^{2}\)
\(P_{4}(x) = 30 + 80\,x + 42\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2527
31418335
43080424156
555250270805660
691630115570013562717
71401372382239201540210711011
820426881058416464109203024308844200
928548602570456448567002646054604329176358
1038582505643016632023562016632057750924058510700910
115061331011434543560082764082328443197011616014850770112778446
126502059221707410381802548260339768025105081019304218790228809901210994920
1381930758390390229028870549051214670612024012687086422252233903903403412481343459650
1410154459067076147367321789287538648610493843353778975217216199455455066566549140154714171655785
Technical details
Node id
n-ae47a8a202baca5a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b7c795684c991b2e

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:   5,    2;
  3:  14,   18,     3;
  4:  30,   80,    42,     4;
  5:  55,  250,   270,    80,     5;
  6:  91,  630,  1155,   700,   135,     6;
  7: 140, 1372,  3822,  3920,  1540,   210,    7;
  8: 204, 2688, 10584, 16464, 10920,  3024,  308,   8;
  9: 285, 4860, 25704, 56448, 56700, 26460, 5460, 432, 9;
%F Formula
The row polynomials satisfy (4*n^3+22*n^2+34*n+12)*P(n,x) = (60*n^3+148*n^2-1226*n-1304-(84*n^3+396*n^2-2226*n-2808)*x-(12*n^3-348*n^2+774*n+1704)*x^2+(36*n^3-100*n^2-226*n+200)*x^3)*P(n-5,x) - (16*n^3+56*n^2-375*n-420-(32*n^3+208*n^2-1016*n-1312)*x+(288*n^2-798*n-1416)*x^2+(32*n^3-176*n^2+48*n+576)*x^3-(16*n^3-40*n^2-109*n+52)*x^4)*d/dx P(n-5,x) - (52*n^3-514*n^2+704*n+1516-(208*n^3+4*n^2-704*n+114)*x-(12*n^3-294*n^2+168*n+600)*x^2+(40*n^3-32*n^2-168*n+118)*x^3)*P(n-4,x) + (64*n^3+176*n^2-504*n-582-(32*n^3+160*n^2-164*n-391)*x-(16*n^3-56*n^2-176*n-9)*x^2-(80*n^2-100*n-217)*x^3-(16*n^3-8*n^2-64*n+35)*x^4)*d/dx P(n-4,x) - (72*n^3+712*n^2+338*n-336+(152*n^3-268*n^2-430*n+330)*x-(12*n^3-24*n^2-102*n+60)*x^2-(4*n^3+4*n^2-14*n+6)*x^3)*P(n-3,x) - (96*n^3+192*n^2+18*n-6+(32*n^3-80*n^2-24*n+150)*x-(32*n^3+272*n^2+71*n-123)*x^2-(32*n^3+32*n^2-58*n+24)*x^3+(3*n-3)*x^4)*d/dx P(n-3,x) + (56*n^3+36*n^2-22*n+8+(16*n^3+76*n^2+100*n+24)*x-(12*n^3+30*n^2+12*n)*x^2)*P(n-2,x) + (64*n^3+80*n^2+48*n+54+(32*n^3-128*n^2-316*n-161)*x-(16*n^3+40*n^2-14*n-24)*x^2+(6*n+3)*x^3)*d/dx P(n-2,x) + (12*n^3+36*n^2+20*n-8+(12*n^3+48*n^2+60*n+24)*x)*P(n-1,x) - (16*n^3+8*n^2-99*n-102+(8*n+16)*x+(3*n+6)*x^2)*d/dx P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 14 + 18*x + 3*x^2, P(4,x) = 30 + 80*x + 42*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[3, x_] := 14 + 18*x + 3*x^2;
P[4, x_] := 30 + 80*x + 42*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((60*n^3 + 148*n^2 - 1226*n - 1304 - (84*n^3 + 396*n^2 - 2226*n - 2808)*x - (12*n^3 - 348*n^2 + 774*n + 1704)*x^2 + (36*n^3 - 100*n^2 - 226*n + 200)*x^3)*P[n - 5, x] - (16*n^3 + 56*n^2 - 375*n - 420 - (32*n^3 + 208*n^2 - 1016*n - 1312)*x + (288*n^2 - 798*n - 1416)*x^2 + (32*n^3 - 176*n^2 + 48*n + 576)*x^3 - (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*D[P[n - 5, x], x] - (52*n^3 - 514*n^2 + 704*n + 1516 - (208*n^3 + 4*n^2 - 704*n + 114)*x - (12*n^3 - 294*n^2 + 168*n + 600)*x^2 + (40*n^3 - 32*n^2 - 168*n + 118)*x^3)*P[n - 4, x] + (64*n^3 + 176*n^2 - 504*n - 582 - (32*n^3 + 160*n^2 - 164*n - 391)*x - (16*n^3 - 56*n^2 - 176*n - 9)*x^2 - (80*n^2 - 100*n - 217)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*D[P[n - 4, x], x] - (72*n^3 + 712*n^2 + 338*n - 336 + (152*n^3 - 268*n^2 - 430*n + 330)*x - (12*n^3 - 24*n^2 - 102*n + 60)*x^2 - (4*n^3 + 4*n^2 - 14*n + 6)*x^3)*P[n - 3, x] - (96*n^3 + 192*n^2 + 18*n - 6 + (32*n^3 - 80*n^2 - 24*n + 150)*x - (32*n^3 + 272*n^2 + 71*n - 123)*x^2 - (32*n^3 + 32*n^2 - 58*n + 24)*x^3 + (3*n - 3)*x^4)*D[P[n - 3, x], x] + (56*n^3 + 36*n^2 - 22*n + 8 + (16*n^3 + 76*n^2 + 100*n + 24)*x - (12*n^3 + 30*n^2 + 12*n)*x^2)*P[n - 2, x] + (64*n^3 + 80*n^2 + 48*n + 54 + (32*n^3 - 128*n^2 - 316*n - 161)*x - (16*n^3 + 40*n^2 - 14*n - 24)*x^2 + (6*n + 3)*x^3)*D[P[n - 2, x], x] + (12*n^3 + 36*n^2 + 20*n - 8 + (12*n^3 + 48*n^2 + 60*n + 24)*x)*P[n - 1, x] - (16*n^3 + 8*n^2 - 99*n - 102 + (8*n + 16)*x + (3*n + 6)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 22*n^2 + 34*n + 12)]];
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] = 5 + 2*x; v[4] = 14 + 18*x + 3*x^2; v[5] = 30 + 80*x + 42*x^2 + 4*x^3; for(n = 5, N, my(i = n + 1); v[i] = ((60*n^3 + 148*n^2 - 1226*n - 1304 - (84*n^3 + 396*n^2 - 2226*n - 2808)*x - (12*n^3 - 348*n^2 + 774*n + 1704)*x^2 + (36*n^3 - 100*n^2 - 226*n + 200)*x^3)*v[i - 5] - (16*n^3 + 56*n^2 - 375*n - 420 - (32*n^3 + 208*n^2 - 1016*n - 1312)*x + (288*n^2 - 798*n - 1416)*x^2 + (32*n^3 - 176*n^2 + 48*n + 576)*x^3 - (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*deriv(v[i - 5], x) - (52*n^3 - 514*n^2 + 704*n + 1516 - (208*n^3 + 4*n^2 - 704*n + 114)*x - (12*n^3 - 294*n^2 + 168*n + 600)*x^2 + (40*n^3 - 32*n^2 - 168*n + 118)*x^3)*v[i - 4] + (64*n^3 + 176*n^2 - 504*n - 582 - (32*n^3 + 160*n^2 - 164*n - 391)*x - (16*n^3 - 56*n^2 - 176*n - 9)*x^2 - (80*n^2 - 100*n - 217)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*deriv(v[i - 4], x) - (72*n^3 + 712*n^2 + 338*n - 336 + (152*n^3 - 268*n^2 - 430*n + 330)*x - (12*n^3 - 24*n^2 - 102*n + 60)*x^2 - (4*n^3 + 4*n^2 - 14*n + 6)*x^3)*v[i - 3] - (96*n^3 + 192*n^2 + 18*n - 6 + (32*n^3 - 80*n^2 - 24*n + 150)*x - (32*n^3 + 272*n^2 + 71*n - 123)*x^2 - (32*n^3 + 32*n^2 - 58*n + 24)*x^3 + (3*n - 3)*x^4)*deriv(v[i - 3], x) + (56*n^3 + 36*n^2 - 22*n + 8 + (16*n^3 + 76*n^2 + 100*n + 24)*x - (12*n^3 + 30*n^2 + 12*n)*x^2)*v[i - 2] + (64*n^3 + 80*n^2 + 48*n + 54 + (32*n^3 - 128*n^2 - 316*n - 161)*x - (16*n^3 + 40*n^2 - 14*n - 24)*x^2 + (6*n + 3)*x^3)*deriv(v[i - 2], x) + (12*n^3 + 36*n^2 + 20*n - 8 + (12*n^3 + 48*n^2 + 60*n + 24)*x)*v[i - 1] - (16*n^3 + 8*n^2 - 99*n - 102 + (8*n + 16)*x + (3*n + 6)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 22*n^2 + 34*n + 12)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.