Real-rooted atlas

A038329 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(21 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 441 + 756\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04417563241521
192612381420412583259319
21944816667928573044898881049762313441
3408410117503290300056402571912011022480188956890224199
4857661214410829089451776601080203040694416240238085568340122243518743761
5180108854110806531246277882232043969746172034026395760174992892484999796928612220032137231006679
63782285936125935674990477807024971213338347137921429108621920979960197888419982941952102852965376110199605765352009260481
77942800465816127303216482210078967422244201579348444854020305908576463031193502082645892534297697196052280322082772548864198359290368208728361158759

All 15 rows

Technical details
Node id
n-bb7cd1e2fa0622c2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8e9037e0ccb64a62

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:             441,              756,               324;
  1:            9261,            23814,             20412,              5832;
  2:          194481,           666792,            857304,            489888,            104976;
  3:         4084101,         17503290,          30005640,          25719120,          11022480,           1889568;
  4:        85766121,        441082908,         945177660,        1080203040,         694416240,         238085568,          34012224;
  5:      1801088541,      10806531246,       27788223204,       39697461720,       34026395760,       17499289248,        4999796928,         612220032;
  6:     37822859361,     259356749904,      778070249712,     1333834713792,     1429108621920,      979960197888,      419982941952,      102852965376,       11019960576;
  7:    794280046581,    6127303216482,    21007896742224,    42015793484448,    54020305908576,    46303119350208,    26458925342976,     9719605228032,     2082772548864,     198359290368;
  8:  16679880978201,  142970408384580,   551457289483380,  1260473804533440,  1890710706800160,  1944731012708736,  1389093580506240,   680372365962240,   218691117630720,   41655450977280,   3570467226624;
  9: 350277500542221, 3302616433683798, 14154070430073420, 36396181105903080, 62393453324405280, 74872143989286336, 64176123419388288, 39291504134319360, 16839216057565440, 4811204587875840, 824777929350144, 64268410079232;
%F Formula
The row polynomials satisfy P(n,x) = (21+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 756*x + 324*x^2.
%t Mathematica
P[0, x_] := 441 + 756*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 18*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] = 441 + 756*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 18*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

Same fingerprint ?

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.