Real-rooted atlas

A125662 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(8 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181220
263192144399
34962280345617287960
4390524000548645529620736158801
53074423658072345610990088294402488323168060
624204722375688571744174182401980288011943936298598463202399
719056322056852894701312241107840366612480332937216167215104358318081260879920
8150030091851782409958610883047860224580639104070509035525329981440229323571242998169625154396001
9118118440164089350010094326272360921070088265883852812575098368012708347904082269831168309586821125159780352501827040100
109299445111435939104099449471088406820911104108856960128019908630282242520349655040218086716211212344774492164127824281606191736422410011386405999
11732143764812439356895295781413491244118688348801350774890496028863988457472439260700999684760886332620836015266856960181108290355205448728051712743008370688199725901079880
125764155667310686464123529055786435632463818993868801599158476569603910140314910726952560726343689058028806471688581940173209605766570521395202608578554757127132880358604889161004482563984506635191601
1345381101573691164764099408431223429836847531276348774418224018249932805018238435354624102102908888678401554188705346355217699171439476736148934737959321609001298741428224370018168602624092727444661862410699320537907279490406802752140
1435728465692157730889705600077483980487059247678670200954882012305191414220861623715351265280141205932262563840245952784213475328327237822140055552330961190589038592250471217627136000137545709581762560518114597048156161198323900245606412839184645488641585823629419851199
Technical details
Node id
n-2021a97cf14f0663 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-ca6d356a3e7f5816

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,         12;
  2:        63,        192,         144;
  3:       496,       2280,        3456,        1728;
  4:      3905,      24000,       54864,       55296,       20736;
  5:     30744,     236580,      723456,     1099008,      829440,       248832;
  6:    242047,    2237568,     8571744,    17418240,    19802880,     11943936,      2985984;
  7:   1905632,   20568528,    94701312,   241107840,   366612480,    332937216,    167215104,    35831808;
  8:  15003009,  185178240,   995861088,  3047860224,  5806391040,   7050903552,   5329981440,  2293235712,   429981696;
  9: 118118440, 1640893500, 10094326272, 36092107008, 82658838528, 125750983680, 127083479040, 82269831168, 30958682112, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (8+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 12*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (8 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (8 + 12*x)*v[i - 1]); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023