Real-rooted atlas

A030528 → basis A013611 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(4 + 4\,n\right)\,P_{n}(x) = \left(56 + 32\,x - 8\,n\right)\,P_{n-1}(x) + \left(45 + 56\,x + 16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
235481699
320542428864981
412003320340815362569720
570252434033456227847680102496309
641125171200294960269056136960368644096954261
72407501170300242424027728641890560768000172032163849455130
8140937578350001896200026086400223052801213440041000967864326553693684519
982506255162700014295240022984128023641625616129536072970240211025923538944262144928256841
1048300000335950000104742000019272960002317318400190203494410790707204177264641055784961572864010485769197472240
1128275312521640875007502170000155471760002139722240020531590144140135260166802309120230090342451642368069206016419430491131561729
1216552656251382440000052748100000121570720000188470592000207026288640165197484032964705648644091255193612286689280247988224030198988816777216902961305721
13969009375087694512500365205300000926590560000159781065600019772606515201806286565376123320841011262915877273623686938624063957893120117272739841308622848671088648946835807050
1456726796875552976000000249584265000069126200000001312405136000018066559692800185945680076801453433107251286690725232643926065676288132881317888032589742080054744055808563714457626843545688648174014939

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-2}(x) + \left(5 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)
Technical details
Node id
n-c543221da2eb63df · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-7bfc29c470808344)
Fingerprint
f6-4285d334207d89bf

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:       6,        4;
  2:      35,       48,        16;
  3:     205,      424,       288,        64;
  4:    1200,     3320,      3408,      1536,       256;
  5:    7025,    24340,     33456,     22784,      7680,      1024;
  6:   41125,   171200,    294960,    269056,    136960,     36864,     4096;
  7:  240750,  1170300,   2424240,   2772864,   1890560,    768000,   172032,    16384;
  8: 1409375,  7835000,  18962000,  26086400,  22305280,  12134400,  4100096,   786432,   65536;
  9: 8250625, 51627000, 142952400, 229841280, 236416256, 161295360, 72970240, 21102592, 3538944, 262144;
%F Formula
The row polynomials satisfy (4*n+4)*P(n,x) = -(8*n-56-32*x)*P(n-1,x) + (45+56*x+16*x^2)*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[(-(8*n - 56 - 32*x)*P[n - 1, x] + (45 + 56*x + 16*x^2)*D[P[n - 1, x], x])/(4*n + 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] = (-(8*n - 56 - 32*x)*v[i - 1] + (45 + 56*x + 16*x^2)*deriv(v[i - 1], x))/(4*n + 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. [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