Real-rooted atlas

A136124 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x - 4\,x^{2} - 2\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(3 + 4\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
212281252
36018814432424
436013681428576803812
5252011016141007840200019237668
62016098208145848100320356006336448406920
7181440964512160725612918085712001424641881610244778520
818144001037088018966096172040968941520279552052057653248230460668640
919958400121337280239739696239821312141366960522103681224921617756161451525120828609120
1023950080015358809603240888192351716160022959568009658051202688147844930560057369603840001126412119446080
1131135104002092445568046739189376543654693123864034240018065762880574813747212554065921855180801774080099123224576189056548800
1243589145600305396421120717122018304885754408704676878466880345657502080122627371328307142922245421917952660633600529070082506752532483133877644800
13653837184000475530289920011673019537920151986415127041236756302496068093708517122645944741440741005517824150435926784219582105602246999040153034752623001611468855019485785600
14104613949440007870019512320020103422130432027430631630438423587761218112013863208488729658223647850368179039619799044074237243648685633428480842600478727351099392431255552152535042457601019991363148800
Technical details
Node id
n-2f3f7d6b662f0ee5 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-13d85ef8fb9be08c

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:        3,         4;
  2:       12,        28,        12;
  3:       60,       188,       144,        32;
  4:      360,      1368,      1428,       576,        80;
  5:     2520,     11016,     14100,      7840,      2000,      192;
  6:    20160,     98208,    145848,    100320,     35600,     6336,      448;
  7:   181440,    964512,   1607256,   1291808,    571200,   142464,    18816,    1024;
  8:  1814400,  10370880,  18966096,  17204096,   8941520,  2795520,   520576,   53248,   2304;
  9: 19958400, 121337280, 239739696, 239821312, 141366960, 52210368, 12249216, 1775616, 145152, 5120;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(4*n+4)*x+4*x^2)*P(n-2,x) + (2*n+3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (4*n + 4)*x + 4*x^2)*P[n - 2, x] + (2*n + 3 + 4*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (4*n + 4)*x + 4*x^2)*v[i - 2] + (2*n + 3 + 4*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.