Real-rooted atlas

A038333 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 56\,x + 28\,n x\right)\,P_{n}(x) = \left(361 + 2128\,x + 2352\,x^{2} + 532\,n x + 784\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 2128\,x + 2352\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0361212823524841
168596064813406487808289379
2130321153641650944326673408307328016507857
3247609936489880161323680316986880291961600103262208912500347
4470458818319692644597724880120455014401664181120011771891712337323212849309176505
5893871739184419853521222994818084005129228807377869632007828307988484486398730241079434280962619349324947
616983563041400454539072309825353913612175592855552280359046016003966342714163234096630349824164074010705923400217985024137294865645473
73226876977798559715772664756859078846083470043963832329588279373747201695611510304768194350792993996814028327915356165814372754391041057845595340807119574711866539

All 15 rows

Technical details
Node id
n-519543e978a17fc6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e740b38b8ddb66d7

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             361,             2128,              2352;
  1:            6859,            60648,            134064,              87808;
  2:          130321,          1536416,           5094432,            6673408,            3073280;
  3:         2476099,         36489880,         161323680,          316986880,          291961600,           103262208;
  4:        47045881,        831969264,        4597724880,        12045501440,        16641811200,         11771891712,          3373232128;
  5:       893871739,      18441985352,      122299481808,       400512922880,       737786963200,        782830798848,        448639873024,        107943428096;
  6:     16983563041,     400454539072,     3098253539136,     12175592855552,     28035904601600,      39663427141632,      34096630349824,      16407401070592,       3400217985024;
  7:    322687697779,    8559715772664,    75685907884608,    347004396383232,    958827937374720,    1695611510304768,    1943507929939968,    1402832791535616,     581437275439104,     105784559534080;
  8:   6131066257801,  180705110756240,  1797540312259440,   9418690758973440,  30362884683532800,   64433237391581184,   92316626672148480,   88846076797255680,   55236541166714880,   20099066311475200,   3258164433649664;
  9: 116490258898219, 3776736814805416, 41742880584691440, 246063296078181120, 906548985551193600, 2244424435806744576, 3858834994895806464, 4642207512656609280, 3848145701281136640, 2100352429549158400, 680956366632779776, 99522113609662464;
%F Formula
The row polynomials satisfy (19+(28*n+56)*x)*P(n,x) = (361+(532*n+2128)*x+(784*n+2352)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 2128*x + 2352*x^2.
%t Mathematica
P[0, x_] := 361 + 2128*x + 2352*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((361 + (532*n + 2128)*x + (784*n + 2352)*x^2)*P[n - 1, x])/(19 + (28*n + 56)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 361 + 2128*x + 2352*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (532*n + 2128)*x + (784*n + 2352)*x^2)*v[i - 1])/(19 + (28*n + 56)*x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.