Real-rooted atlas

A116412 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(16 + 16\,n\right)\,P_{n}(x) = \left(512 + 256\,x - 304\,n - 128\,n x + 48\,n^{2}\right)\,P_{n-1}(x) + \left(1092 + 1248\,x + 320\,x^{2} - 276\,n - 288\,n x - 64\,n x^{2}\right)\,P_{n-1}'(x) + \left(345 + 636\,x + 368\,x^{2} + 64\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2616416141
3467744384641659
4357476406000204825619518
527353733647745640192102401024229629
62093416754568962086254082419204915240962701581
71602152604178896560168473472440704013578242293761638431784052

All 15 rows

Technical details
Node id
n-f6fbd9e8c5ec6b97 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e06a50cf7d5abe3a

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:        8,         4;
  2:       61,        64,        16;
  3:      467,       744,       384,         64;
  4:     3574,      7640,      6000,       2048,       256;
  5:    27353,     73364,     77456,      40192,     10240,      1024;
  6:   209341,    675456,    896208,     625408,    241920,     49152,      4096;
  7:  1602152,   6041788,   9656016,    8473472,   4407040,   1357824,    229376,    16384;
  8: 12261769,  52915768,  98942128,  104650240,  68454400,  28346368,   7254016,  1048576,   65536;
  9: 93843143, 456075000, 976705200, 1209311616, 953710848, 496659456, 170741760, 37355520, 4718592, 262144;
%F Formula
The row polynomials satisfy (16*n+16)*P(n,x) = (48*n^2-304*n+512-(128*n-256)*x)*P(n-1,x) - (276*n-1092+(288*n-1248)*x+(64*n-320)*x^2)*d/dx P(n-1,x) + (345+636*x+368*x^2+64*x^3)*d^2/dx^2 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[((48*n^2 - 304*n + 512 - (128*n - 256)*x)*P[n - 1, x] - (276*n - 1092 + (288*n - 1248)*x + (64*n - 320)*x^2)*D[P[n - 1, x], x] + (345 + 636*x + 368*x^2 + 64*x^3)*D[P[n - 1, x], {x, 2}])/(16*n + 16)]];
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] = ((48*n^2 - 304*n + 512 - (128*n - 256)*x)*v[i - 1] - (276*n - 1092 + (288*n - 1248)*x + (64*n - 320)*x^2)*deriv(v[i - 1], x) + (345 + 636*x + 368*x^2 + 64*x^3)*deriv(deriv(v[i - 1], x), x))/(16*n + 16)); 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

Candidate equalities ?

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