Real-rooted atlas

A099040 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(24 + 16\,x - 4\,n\right)\,P_{n-1}(x) + \left(35 + 48\,x + 16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110818
211016864342
31200272020485126480
4131003936044224220164096122796
5143000535200799360595456221184327682326968
61561000699440013032320129233927192576212992026214444095752
7170400008892800019855360024584192018228838480936960199229442097152835608960
8186010000110803200028832384004280340480396493209623465164808663859201824522241677721615834684816
9203050000013594000000403936000006991616000077681664000574522982402828271616089359646721644167168134217728300065287968
1022165100000164752400000550384640000108817971200014100379648001251160915968769881538560324376985600895567790081462973235210737418245686199510112
112419560000001977028800000733455360000016307184640000241419632640002498788917248018450548260864971844354048035785130639368772470702081288490188808589934592107752766365440
12264121100000023530780800000959836320000002370331494400003946829271040004668065107148804021166987018242541916436561921170232626380803826332676915284342420275201125281431552687194767362041901385759936
13288316700000002781434320000001237244332800000335994882560000062149715763200008268543121817600814002766872576060036388476682243317298454593536135621994400972839875550117888079843442032640975816569651254975581388838693774738256768
1431472881000000032685251760000001574567335040000046635643468800000948724008345600001402307323527168001553042407345356801309154598526648328440586110291148841419406445641728152277635982950404067205180293120746018639446016841126395248644398046511104733242170232300672

Other recurrences

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

\(P_{n}(x) = \left(10 + 8\,x\right)\,P_{n-2}(x) + \left(10 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 10 + 8\,x\)
Technical details
Node id
n-e42c9f6d99a2eabc · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-021d8139b495357a)
Fingerprint
f6-5904ea7c77900e95

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:         10,           8;
  2:        110,         168,          64;
  3:       1200,        2720,        2048,         512;
  4:      13100,       39360,       44224,       22016,        4096;
  5:     143000,      535200,      799360,      595456,      221184,       32768;
  6:    1561000,     6994400,    13032320,    12923392,     7192576,     2129920,      262144;
  7:   17040000,    88928000,   198553600,   245841920,   182288384,    80936960,    19922944,    2097152;
  8:  186010000,  1108032000,  2883238400,  4280340480,  3964932096,  2346516480,   866385920,  182452224,   16777216;
  9: 2030500000, 13594000000, 40393600000, 69916160000, 77681664000, 57452298240, 28282716160, 8935964672, 1644167168, 134217728;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = -(4*n-24-16*x)*P(n-1,x) + (35+48*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[(-(4*n - 24 - 16*x)*P[n - 1, x] + (35 + 48*x + 16*x^2)*D[P[n - 1, x], x])/(2*n)]];
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] = (-(4*n - 24 - 16*x)*v[i - 1] + (35 + 48*x + 16*x^2)*deriv(v[i - 1], x))/(2*n)); 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