Real-rooted atlas

basis-map coefficients of A021012 → basis A008277 → x ↦ x/(x+1)

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 5\,x + 6\,x^{2} + 2\,x^{3} - n - 6\,n x - 8\,n x^{2} - 3\,n x^{3} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-1-2-3
32101022
4-6-60-138-90-294
5244561824261612486168
6-120-4320-26880-63720-65280-24360-184680
7720496804543201586880262152020707206314407415280
8-5040-675360-8814960-42693840-101248560-127048320-81209520-20865600-382561200
940320106041601950681601263265920403889472071620819207199700480385035840085264704024572661120
10-362880-188697600-4882187520-41298284160-171216581760-402605562240-566015869440-472633418880-216695606400-42085008000-1917621578880
11362880037485504001369980864001490369932800780609312000023390602502400431384486400005000724449280035606467814400142584369984002462394816000178300808582400
12-39916800-82148774400-4273133356800-59177254752000-384491151475200-1429021280332800-3321051654412800-5012246117894400-4927426709990400-3051558169228800-1083035550643200-168193308729600-19440556519507200
1347900160019672595712001469270347776002574262373760000204775895149056009259462191006720026381933761075200049784395469276160063363375248785920053950301028771840029514161340195840093943133080473600132417295540992002452921899687705600
14-6227020800-51086478643200-5525484636672000-122120016604185600-1178081160512256000-6388774483867776000-21877890413028249600-50208963472110643200-79445126351464473600-87198966976426905600-65364608657034086400-31982921101587456000-9219299106745728000-1188734048799897600-354181062365523993600
15871782912001429462440806400225632325113395200628864090702848000073066355439517440000470080845578792448000190615219147807534080052141516324655966208009974829125636402227200135514612896449859072001305048965227057290240087259292851730807808003857968723476759091200101538252014805365760012056396275353830400057966591286847135692800
Technical details
Node id
n-36699036e6968a66 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-629d13292f83019a

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 -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:      -1,         -2;
   3:       2,         10,          10;
   4:      -6,        -60,        -138,          -90;
   5:      24,        456,        1824,         2616,          1248;
   6:    -120,      -4320,      -26880,       -63720,        -65280,        -24360;
   7:     720,      49680,      454320,      1586880,       2621520,       2070720,        631440;
   8:   -5040,    -675360,    -8814960,    -42693840,    -101248560,    -127048320,     -81209520,     -20865600;
   9:   40320,   10604160,   195068160,   1263265920,    4038894720,    7162081920,    7199700480,    3850358400,     852647040;
  10: -362880, -188697600, -4882187520, -41298284160, -171216581760, -402605562240, -566015869440, -472633418880, -216695606400, -42085008000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-1-(n^2-6*n+5)*x-(2*n^2-8*n+6)*x^2-(n^2-3*n+2)*x^3)*P(n-1,x) - ((n-1)*x+(3*n-3)*x^2+(3*n-3)*x^3+(n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 - (n^2 - 6*n + 5)*x - (2*n^2 - 8*n + 6)*x^2 - (n^2 - 3*n + 2)*x^3)*P[n - 1, x] - ((n - 1)*x + (3*n - 3)*x^2 + (3*n - 3)*x^3 + (n - 1)*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - 1 - (n^2 - 6*n + 5)*x - (2*n^2 - 8*n + 6)*x^2 - (n^2 - 3*n + 2)*x^3)*v[i - 1] - ((n - 1)*x + (3*n - 3)*x^2 + (3*n - 3)*x^3 + (n - 1)*x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741