Real-rooted atlas

A121574 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(50\,n + 112\,n x - 80\,n x^{2} + 64\,n x^{3} + 32\,n x^{4} + 60\,n^{2} + 8\,n^{2} x + 112\,n^{2} x^{2} + 32\,n^{2} x^{3} - 24\,n^{3} + 48\,n^{3} x\right)\,P_{n}(x) = \left(-118 - 380\,x - 16\,x^{2} + 736\,x^{3} + 416\,x^{4} + 64\,x^{5} + 524\,n + 1424\,n x + 864\,n x^{2} - 64\,n x^{3} - 64\,n x^{4} + 216\,n^{2} + 160\,n^{2} x - 352\,n^{2} x^{2} - 128\,n^{2} x^{3} - 192\,n^{3} - 192\,n^{3} x\right)\,P_{n-1}(x) + \left(39 + 424\,x + 1612\,x^{2} + 2752\,x^{3} + 2128\,x^{4} + 640\,x^{5} + 64\,x^{6} + 234\,n + 1244\,n x + 2368\,n x^{2} + 1920\,n x^{3} + 608\,n x^{4} + 64\,n x^{5} + 156\,n^{2} + 552\,n^{2} x + 528\,n^{2} x^{2} + 96\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22422450
3103158688337
4419918664184162201
516444716490822804643214044
66289223803069620536697611206488061
72360910049017190015192874224198082624128544711

All 15 rows

Technical details
Node id
n-b5909c1bdd3da339 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f1372994cd6eaf73

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:      5,       2;
  2:     24,      22,       4;
  3:    103,     158,      68,       8;
  4:    419,     918,     664,     184,      16;
  5:   1644,    4716,    4908,    2280,     464,      32;
  6:   6289,   22380,   30696,   20536,    6976,    1120,     64;
  7:  23609,  100490,  171900,  151928,   74224,   19808,   2624,    128;
  8:  87348,  433210,  890224,  983880,  635216,  243296,  53376,   6016,   256;
  9: 319447, 1810154, 4349612, 5789352, 4679008, 2366848, 743616, 138368, 13568, 512;
%F Formula
The row polynomials satisfy -(24*n^3-60*n^2-50*n-(48*n^3+8*n^2+112*n)*x-(112*n^2-80*n)*x^2-(32*n^2+64*n)*x^3-32*n*x^4)*P(n,x) = -(192*n^3-216*n^2-524*n+118+(192*n^3-160*n^2-1424*n+380)*x+(352*n^2-864*n+16)*x^2+(128*n^2+64*n-736)*x^3+(64*n-416)*x^4-64*x^5)*P(n-1,x) + (156*n^2+234*n+39+(552*n^2+1244*n+424)*x+(528*n^2+2368*n+1612)*x^2+(96*n^2+1920*n+2752)*x^3+(608*n+2128)*x^4+(64*n+640)*x^5+64*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[(-(192*n^3 - 216*n^2 - 524*n + 118 + (192*n^3 - 160*n^2 - 1424*n + 380)*x + (352*n^2 - 864*n + 16)*x^2 + (128*n^2 + 64*n - 736)*x^3 + (64*n - 416)*x^4 - 64*x^5)*P[n - 1, x] + (156*n^2 + 234*n + 39 + (552*n^2 + 1244*n + 424)*x + (528*n^2 + 2368*n + 1612)*x^2 + (96*n^2 + 1920*n + 2752)*x^3 + (608*n + 2128)*x^4 + (64*n + 640)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(-24*n^3 + 60*n^2 + 50*n + (48*n^3 + 8*n^2 + 112*n)*x + (112*n^2 - 80*n)*x^2 + (32*n^2 + 64*n)*x^3 + 32*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] = (-(192*n^3 - 216*n^2 - 524*n + 118 + (192*n^3 - 160*n^2 - 1424*n + 380)*x + (352*n^2 - 864*n + 16)*x^2 + (128*n^2 + 64*n - 736)*x^3 + (64*n - 416)*x^4 - 64*x^5)*v[i - 1] + (156*n^2 + 234*n + 39 + (552*n^2 + 1244*n + 424)*x + (528*n^2 + 2368*n + 1612)*x^2 + (96*n^2 + 1920*n + 2752)*x^3 + (608*n + 2128)*x^4 + (64*n + 640)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(-24*n^3 + 60*n^2 + 50*n + (48*n^3 + 8*n^2 + 112*n)*x + (112*n^2 - 80*n)*x^2 + (32*n^2 + 64*n)*x^3 + 32*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.