Real-rooted atlas

A119308 → θ+1

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)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 3 + 10\,x + 3\,x^{2}\)
\(P_{3}(x) = 4 + 28\,x + 27\,x^{2} + 4\,x^{3}\)
\(P_{4}(x) = 5 + 60\,x + 120\,x^{2} + 56\,x^{3} + 5\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2310316
342827463
4560120565246
561103753601006957
67182945154087516273718
782802058509649001848245814443
8940840321411220580131043528352956134
910570729034272705606804030870624048610218348
10117701237575240207900282744194040660001039565011850136
111210121996515246054450099316896049849368013068016500847123313334
1213130030888289432129772530579123963960286915211467172431002516810801312926460
1314163846137520520286286084658861417115713741728772972224724704294293712813521450480055
141520306688589434859209152147145045090045564392404251347119129110501000572618053235166615197318610
Technical details
Node id
n-cbfb640e3f6accdd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a0c3ea8a600f0697

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:  2,   2;
  2:  3,  10,    3;
  3:  4,  28,   27,     4;
  4:  5,  60,  120,    56,     5;
  5:  6, 110,  375,   360,   100,     6;
  6:  7, 182,  945,  1540,   875,   162,     7;
  7:  8, 280, 2058,  5096,  4900,  1848,   245,    8;
  8:  9, 408, 4032, 14112, 20580, 13104,  3528,  352,   9;
  9: 10, 570, 7290, 34272, 70560, 68040, 30870, 6240, 486, 10;
%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) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 10*x + 3*x^2, P(3,x) = 4 + 28*x + 27*x^2 + 4*x^3, P(4,x) = 5 + 60*x + 120*x^2 + 56*x^3 + 5*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 3 + 10*x + 3*x^2;
P[3, x_] := 4 + 28*x + 27*x^2 + 4*x^3;
P[4, x_] := 5 + 60*x + 120*x^2 + 56*x^3 + 5*x^4;
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] = 1; v[2] = 2 + 2*x; v[3] = 3 + 10*x + 3*x^2; v[4] = 4 + 28*x + 27*x^2 + 4*x^3; v[5] = 5 + 60*x + 120*x^2 + 56*x^3 + 5*x^4; 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.