Real-rooted atlas

A110441 → basis A013611 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2323264
3176384192752
48322880307210247808
53632174082918420480512075824
6151049292821504023552012288024576706048
7608004577281359360204800016588806881281146886387584
823910421314567753728149708801662976010665984367001652428856585216
99239689523200410849289714073613805568012111052864225280188743682359296493297984
103522048412216322059714565777408001004666880112577740881474355236805017694371840104857604246551552
11132794881740165129890457603215196160663126016090832896008384151552515899392020289945604613734404613734436185738496
12496220167198586884588339200169802137604059168768065911652352742958039045822742528031142707200108422758402214592512201326592305765505024
131840432642928377856206944174088596160512023400529920044032013107258800262348856222967398438253625344018085838848056472109056104689827848724152322565534320384
146782894081174614016091171258368420256219136128464584704027526826557444258246164480482047583846439978931322882401910128640101723616051228789702656048855252992375809638421397452210176
Technical details
Node id
n-af6363abf4915fae · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b099b1bffeb411af

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      4;
  2:     32,      32;
  3:    176,     384,      192;
  4:    832,    2880,     3072,     1024;
  5:   3632,   17408,    29184,    20480,      5120;
  6:  15104,   92928,   215040,   235520,    122880,     24576;
  7:  60800,  457728,  1359360,  2048000,   1658880,    688128,   114688;
  8: 239104, 2131456,  7753728, 14970880,  16629760,  10665984,  3670016,   524288;
  9: 923968, 9523200, 41084928, 97140736, 138055680, 121110528, 64225280, 18874368, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+4+(2*n+4)*x)*P(n-1,x) + (1+4*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (2*n + 4)*x)*P[n - 1, x] + (1 + 4*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 4 + (2*n + 4)*x)*v[i - 1] + (1 + 4*x + 2*x^2)*deriv(v[i - 1], x))/(n - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023