Real-rooted atlas

A119308 → D+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(-884 + 1496\,x - 288\,x^{2} - 376\,x^{3} + 52\,x^{4} - 851\,n + 1210\,n x + 24\,n x^{2} - 274\,n x^{3} - 109\,n x^{4} + 92\,n^{2} - 188\,n^{2} x + 60\,n^{2} x^{2} + 76\,n^{2} x^{3} - 40\,n^{2} x^{4} + 44\,n^{3} - 52\,n^{3} x - 12\,n^{3} x^{2} + 4\,n^{3} x^{3} + 16\,n^{3} x^{4}\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(-2098 + 505\,x + 609\,x^{2} + 99\,x^{3} - 35\,x^{4} - 1208\,n - 540\,n x + 344\,n x^{2} + 268\,n x^{3} + 64\,n x^{4} + 690\,n^{2} - 156\,n^{2} x - 238\,n^{2} x^{2} - 48\,n^{2} x^{3} + 8\,n^{2} x^{4} + 12\,n^{3} + 176\,n^{3} x - 4\,n^{3} x^{2} - 40\,n^{3} x^{3} - 16\,n^{3} x^{4}\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(342 - 480\,x - 63\,x^{2} + 30\,x^{3} + 3\,x^{4} - 356\,n + 454\,n x - 31\,n x^{2} - 72\,n x^{3} - 3\,n x^{4} - 904\,n^{2} + 348\,n^{2} x + 248\,n^{2} x^{2} + 36\,n^{2} x^{3} - 168\,n^{3} - 184\,n^{3} x + 44\,n^{3} x^{2} + 36\,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(62 - 137\,x + 24\,x^{2} + 3\,x^{3} + 26\,n - 216\,n x + 2\,n x^{2} + 6\,n x^{3} + 116\,n^{2} - 52\,n^{2} x - 70\,n^{2} x^{2} + 120\,n^{3} + 48\,n^{3} x - 28\,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(94 + 8\,x - 6\,x^{2} + 119\,n + 52\,n x - 3\,n x^{2} + 28\,n^{2} + 48\,n^{2} x - 4\,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)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
\(P_{2}(x) = 8 + 7\,x + x^{2}\)
\(P_{3}(x) = 18 + 32\,x + 12\,x^{2} + x^{3}\)
\(P_{4}(x) = 35 + 110\,x + 82\,x^{2} + 18\,x^{3} + x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
287116
3183212163
43511082181246
561305395170251957
698721147010853103313718
7148151245085194252051842114443

All 15 rows

Technical details
Node id
n-13054822f21023a5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a155e4872033def2

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:   3,    1;
  2:   8,    7,     1;
  3:  18,   32,    12,     1;
  4:  35,  110,    82,    18,     1;
  5:  61,  305,   395,   170,    25,     1;
  6:  98,  721,  1470,  1085,   310,    33,    1;
  7: 148, 1512,  4508,  5194,  2520,   518,   42,    1;
  8: 213, 2892, 11928, 19992, 15036,  5208,  812,   52,  1;
  9: 295, 5145, 28134, 65016, 70812, 37800, 9870, 1212, 63, 1;
%F Formula
The row polynomials satisfy (4*n^3+22*n^2+34*n+12)*P(n,x) = (44*n^3+92*n^2-851*n-884-(52*n^3+188*n^2-1210*n-1496)*x-(12*n^3-60*n^2-24*n+288)*x^2+(4*n^3+76*n^2-274*n-376)*x^3+(16*n^3-40*n^2-109*n+52)*x^4)*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) + (12*n^3+690*n^2-1208*n-2098+(176*n^3-156*n^2-540*n+505)*x-(4*n^3+238*n^2-344*n-609)*x^2-(40*n^3+48*n^2-268*n-99)*x^3-(16*n^3-8*n^2-64*n+35)*x^4)*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) - (168*n^3+904*n^2+356*n-342+(184*n^3-348*n^2-454*n+480)*x-(44*n^3+248*n^2-31*n-63)*x^2-(36*n^3+36*n^2-72*n+30)*x^3+(3*n-3)*x^4)*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) + (120*n^3+116*n^2+26*n+62+(48*n^3-52*n^2-216*n-137)*x-(28*n^3+70*n^2-2*n-24)*x^2+(6*n+3)*x^3)*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) - (4*n^3-28*n^2-119*n-94-(12*n^3+48*n^2+52*n+8)*x+(3*n+6)*x^2)*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) = 3 + x, P(2,x) = 8 + 7*x + x^2, P(3,x) = 18 + 32*x + 12*x^2 + x^3, P(4,x) = 35 + 110*x + 82*x^2 + 18*x^3 + x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 8 + 7*x + x^2;
P[3, x_] := 18 + 32*x + 12*x^2 + x^3;
P[4, x_] := 35 + 110*x + 82*x^2 + 18*x^3 + x^4;
P[n_, x_] := P[n, x] = Expand[Together[((44*n^3 + 92*n^2 - 851*n - 884 - (52*n^3 + 188*n^2 - 1210*n - 1496)*x - (12*n^3 - 60*n^2 - 24*n + 288)*x^2 + (4*n^3 + 76*n^2 - 274*n - 376)*x^3 + (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*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] + (12*n^3 + 690*n^2 - 1208*n - 2098 + (176*n^3 - 156*n^2 - 540*n + 505)*x - (4*n^3 + 238*n^2 - 344*n - 609)*x^2 - (40*n^3 + 48*n^2 - 268*n - 99)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*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] - (168*n^3 + 904*n^2 + 356*n - 342 + (184*n^3 - 348*n^2 - 454*n + 480)*x - (44*n^3 + 248*n^2 - 31*n - 63)*x^2 - (36*n^3 + 36*n^2 - 72*n + 30)*x^3 + (3*n - 3)*x^4)*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] + (120*n^3 + 116*n^2 + 26*n + 62 + (48*n^3 - 52*n^2 - 216*n - 137)*x - (28*n^3 + 70*n^2 - 2*n - 24)*x^2 + (6*n + 3)*x^3)*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] - (4*n^3 - 28*n^2 - 119*n - 94 - (12*n^3 + 48*n^2 + 52*n + 8)*x + (3*n + 6)*x^2)*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] = 3 + x; v[3] = 8 + 7*x + x^2; v[4] = 18 + 32*x + 12*x^2 + x^3; v[5] = 35 + 110*x + 82*x^2 + 18*x^3 + x^4; for(n = 5, N, my(i = n + 1); v[i] = ((44*n^3 + 92*n^2 - 851*n - 884 - (52*n^3 + 188*n^2 - 1210*n - 1496)*x - (12*n^3 - 60*n^2 - 24*n + 288)*x^2 + (4*n^3 + 76*n^2 - 274*n - 376)*x^3 + (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*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) + (12*n^3 + 690*n^2 - 1208*n - 2098 + (176*n^3 - 156*n^2 - 540*n + 505)*x - (4*n^3 + 238*n^2 - 344*n - 609)*x^2 - (40*n^3 + 48*n^2 - 268*n - 99)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*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) - (168*n^3 + 904*n^2 + 356*n - 342 + (184*n^3 - 348*n^2 - 454*n + 480)*x - (44*n^3 + 248*n^2 - 31*n - 63)*x^2 - (36*n^3 + 36*n^2 - 72*n + 30)*x^3 + (3*n - 3)*x^4)*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) + (120*n^3 + 116*n^2 + 26*n + 62 + (48*n^3 - 52*n^2 - 216*n - 137)*x - (28*n^3 + 70*n^2 - 2*n - 24)*x^2 + (6*n + 3)*x^3)*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) - (4*n^3 - 28*n^2 - 119*n - 94 - (12*n^3 + 48*n^2 + 52*n + 8)*x + (3*n + 6)*x^2)*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.