Real-rooted atlas

A063007 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(-2 + n\right)\,P_{n-2}(x) + \left(27 + 36\,x - 18\,n - 24\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 9 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
291221
3121324216661
4180972729720432023121
52840115228030564027216090720849201
645864930745808232840110073607348320195955232081301
7754404160693948203227920362517120363333600193995648431101441234422421
812570012911799449764742360784105788592001414551600011337967872504388684896074035248114976161
9211458864122686717168106398275760284904164160476411997600509432571648340182146304129699947520216166579201893447066721
1035836273449432556027740231878751192072456616108801454425788816019448947469952173257513228809914840432640330734866176048997757952075063964778901
1161089714620181933570635404941778543812017651275507008041347912381776066372759227174473940018600448056445417160704028259456898816083786166097920111714888130562993348092318101
121046374526340915437863687039210346817775179604158339638412960111348478273032002085879678788505627893884626736128266283215365939201778361375428352079131379092480002111411385667584255928652808192119937806278590321
131799396167431212896187049764424213536696735522169536723628672240028733891417431200061530732765940838496024405242871705611003804765972029449189596875814784005454479412974592002184137777840578565297723113129574458863590145884164824752800295302161
143104680678772409541362636598066924354627392421815602139525306993634560716367756933148320017261193647246605056307932602675971553284117994485419589632041282051486149393920306399027239585280001636554965030944358459569953227634769921324430778282393600135839054182809600194735074344591580821
155372129928028820110088127269720249408791428382562236320471253430744423308801735888887806587113604648216138203395197449330647010037453900801426388427148236963840166870672628251527936014866993014426756403209929512620340403650564820928032947912704001608447600736284672003300889016642273280031437038253735936007882290383355684814101
Technical details
Node id
n-421f5562485e7f0c · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-639e3b5d50844ba9

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           1;
   2:           9,           12;
   3:         121,          324,           216;
   4:        1809,         7272,          9720,          4320;
   5:       28401,       152280,        305640,        272160,          90720;
   6:      458649,      3074580,       8232840,      11007360,        7348320,        1959552;
   7:     7544041,     60693948,     203227920,     362517120,      363333600,      193995648,       43110144;
   8:   125700129,   1179944976,    4742360784,   10578859200,    14145516000,    11337967872,     5043886848,     960740352;
   9:  2114588641,  22686717168,  106398275760,  284904164160,   476411997600,   509432571648,   340182146304,  129699947520,   21616657920;
  10: 35836273449, 432556027740, 2318787511920, 7245661610880, 14544257888160, 19448947469952, 17325751322880, 9914840432640, 3307348661760, 489977579520;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - (18*n-27+(24*n-36)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 9 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 9 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*P[n - 2, x] - (18*n - 27 + (24*n - 36)*x)*P[n - 1, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 9 + 12*x; for(n = 3, N, my(i = n); v[i] = ((n - 2)*v[i - 2] - (18*n - 27 + (24*n - 36)*x)*v[i - 1])/(-n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023