Real-rooted atlas

A081579 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(10\,n + 10\,n x + 9\,n x^{2} + 18\,n x^{3} + n x^{4} + 10\,n^{2} + 15\,n^{2} x + 19\,n^{2} x^{2} + n^{2} x^{3} + 5\,n^{3} x\right)\,P_{n}(x) = \left(-80 + 20\,x + 308\,x^{2} + 237\,x^{3} + 30\,x^{4} + x^{5} + 180\,n + 290\,n x - 28\,n x^{2} - 143\,n x^{3} - 8\,n x^{4} + 10\,n^{2} - 155\,n^{2} x - 166\,n^{2} x^{2} - 9\,n^{2} x^{3} - 50\,n^{3} - 45\,n^{3} x\right)\,P_{n-1}(x) + \left(200 + 980\,x + 1770\,x^{2} + 1439\,x^{3} + 489\,x^{4} + 41\,x^{5} + x^{6} + 300\,n + 980\,n x + 1095\,n x^{2} + 454\,n x^{3} + 40\,n x^{4} + n x^{5} + 100\,n^{2} + 200\,n^{2} x + 105\,n^{2} x^{2} + 5\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
21610127
36064171142
4240360160241785
5896176011603043114152
6332880967120278449638121863
7120963520039056207845556736451113474
8435201475841982721347204937698001024521584349
91548806003209491207892483727521016801584013605912985260
10546816238592043443204286208249651288179218864024000174466115156019
11191692893020161918464021946368152689926686976185571232336034604217673176521846
12668057635696640822927361072199688698291245733632157862403567872521104479762656801384532393
13231628801351639043445739525041520644680747522882803201198894083375206463853767992966444031848711924301728
14799539205059788801413754880229634867224040161281701322240831704320282734080666411521078332811775208432037609419594503295
Technical details
Node id
n-a6a372c6186eade0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-eb9f6c64faefc4c0

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:     16,     10,      1;
  3:     60,     64,     17,      1;
  4:    240,    360,    160,     24,      1;
  5:    896,   1760,   1160,    304,     31,      1;
  6:   3328,   8096,   7120,   2784,    496,     38,     1;
  7:  12096,  35200,  39056,  20784,   5556,    736,    45,    1;
  8:  43520, 147584, 198272, 134720,  49376,   9800,  1024,   52,  1;
  9: 154880, 600320, 949120, 789248, 372752, 101680, 15840, 1360, 59, 1;
%F Formula
The row polynomials satisfy (10*n^2+10*n+(5*n^3+15*n^2+10*n)*x+(19*n^2+9*n)*x^2+(n^2+18*n)*x^3+n*x^4)*P(n,x) = -(50*n^3-10*n^2-180*n+80+(45*n^3+155*n^2-290*n-20)*x+(166*n^2+28*n-308)*x^2+(9*n^2+143*n-237)*x^3+(8*n-30)*x^4-x^5)*P(n-1,x) + (100*n^2+300*n+200+(200*n^2+980*n+980)*x+(105*n^2+1095*n+1770)*x^2+(5*n^2+454*n+1439)*x^3+(40*n+489)*x^4+(n+41)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(50*n^3 - 10*n^2 - 180*n + 80 + (45*n^3 + 155*n^2 - 290*n - 20)*x + (166*n^2 + 28*n - 308)*x^2 + (9*n^2 + 143*n - 237)*x^3 + (8*n - 30)*x^4 - x^5)*P[n - 1, x] + (100*n^2 + 300*n + 200 + (200*n^2 + 980*n + 980)*x + (105*n^2 + 1095*n + 1770)*x^2 + (5*n^2 + 454*n + 1439)*x^3 + (40*n + 489)*x^4 + (n + 41)*x^5 + x^6)*D[P[n - 1, x], x])/(10*n^2 + 10*n + (5*n^3 + 15*n^2 + 10*n)*x + (19*n^2 + 9*n)*x^2 + (n^2 + 18*n)*x^3 + n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(50*n^3 - 10*n^2 - 180*n + 80 + (45*n^3 + 155*n^2 - 290*n - 20)*x + (166*n^2 + 28*n - 308)*x^2 + (9*n^2 + 143*n - 237)*x^3 + (8*n - 30)*x^4 - x^5)*v[i - 1] + (100*n^2 + 300*n + 200 + (200*n^2 + 980*n + 980)*x + (105*n^2 + 1095*n + 1770)*x^2 + (5*n^2 + 454*n + 1439)*x^3 + (40*n + 489)*x^4 + (n + 41)*x^5 + x^6)*deriv(v[i - 1], x))/(10*n^2 + 10*n + (5*n^3 + 15*n^2 + 10*n)*x + (19*n^2 + 9*n)*x^2 + (n^2 + 18*n)*x^3 + n*x^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.