Real-rooted atlas

A081578 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8\,n x^{2} + 11\,n x^{3} + 2\,n x^{4} + 4\,n^{2} + 4\,n^{2} x + 14\,n^{2} x^{2} + 2\,n^{2} x^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-28 - 92\,x - 22\,x^{2} + 121\,x^{3} + 44\,x^{4} + 4\,x^{5} + 24\,n + 132\,n x + 144\,n x^{2} - 43\,n x^{3} - 10\,n x^{4} + 36\,n^{2} + 28\,n^{2} x - 82\,n^{2} x^{2} - 14\,n^{2} x^{3} - 16\,n^{3} - 28\,n^{3} x\right)\,P_{n-1}(x) + \left(28\,x + 172\,x^{2} + 367\,x^{3} + 301\,x^{4} + 64\,x^{5} + 4\,x^{6} + 16\,n + 120\,n x + 300\,n x^{2} + 278\,n x^{3} + 62\,n x^{4} + 4\,n x^{5} + 16\,n^{2} + 64\,n^{2} x + 68\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
22122447
380156728316
4305868720200162109
510924142524026165123213634
63829181863123623864830412486486731
7131207503216401617259291648241922944128543672

All 15 rows

Technical details
Node id
n-eae72f35531ad048 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0aeec668383753c4

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:      4,       2;
  2:     21,      22,       4;
  3:     80,     156,      72,       8;
  4:    305,     868,     720,     200,      16;
  5:   1092,    4142,    5240,    2616,     512,      32;
  6:   3829,   18186,   31236,   23864,    8304,    1248,     64;
  7:  13120,   75032,  164016,  172592,   91648,   24192,   2944,    128;
  8:  44289,  296488,  789184, 1067952,  784736,  314240,  66432,   6784,   256;
  9: 147620, 1132986, 3561360, 5921040, 5632512, 3120960, 994176, 174720, 15360, 512;
%F Formula
The row polynomials satisfy (4*n^2+(4*n^3+4*n^2)*x+(14*n^2-8*n)*x^2+(2*n^2+11*n)*x^3+2*n*x^4)*P(n,x) = -(16*n^3-36*n^2-24*n+28+(28*n^3-28*n^2-132*n+92)*x+(82*n^2-144*n+22)*x^2+(14*n^2+43*n-121)*x^3+(10*n-44)*x^4-4*x^5)*P(n-1,x) + (16*n^2+16*n+(64*n^2+120*n+28)*x+(68*n^2+300*n+172)*x^2+(8*n^2+278*n+367)*x^3+(62*n+301)*x^4+(4*n+64)*x^5+4*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[(-(16*n^3 - 36*n^2 - 24*n + 28 + (28*n^3 - 28*n^2 - 132*n + 92)*x + (82*n^2 - 144*n + 22)*x^2 + (14*n^2 + 43*n - 121)*x^3 + (10*n - 44)*x^4 - 4*x^5)*P[n - 1, x] + (16*n^2 + 16*n + (64*n^2 + 120*n + 28)*x + (68*n^2 + 300*n + 172)*x^2 + (8*n^2 + 278*n + 367)*x^3 + (62*n + 301)*x^4 + (4*n + 64)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(4*n^2 + (4*n^3 + 4*n^2)*x + (14*n^2 - 8*n)*x^2 + (2*n^2 + 11*n)*x^3 + 2*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] = (-(16*n^3 - 36*n^2 - 24*n + 28 + (28*n^3 - 28*n^2 - 132*n + 92)*x + (82*n^2 - 144*n + 22)*x^2 + (14*n^2 + 43*n - 121)*x^3 + (10*n - 44)*x^4 - 4*x^5)*v[i - 1] + (16*n^2 + 16*n + (64*n^2 + 120*n + 28)*x + (68*n^2 + 300*n + 172)*x^2 + (8*n^2 + 278*n + 367)*x^3 + (62*n + 301)*x^4 + (4*n + 64)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(4*n^2 + (4*n^3 + 4*n^2)*x + (14*n^2 - 8*n)*x^2 + (2*n^2 + 11*n)*x^3 + 2*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.