Real-rooted atlas

A139548 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - x + 2\,n x\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} + 16\,n x + 32\,n x^{2} + 16\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18322464
264512115210243203072
35126144230404096038400184323584131072
440966553634406491750414336001376256802816262144368645242880
53276865536044236801572864034406400495452164816896031457280132710403276800360448201326592
626214462914565190451223068672064880640012457082881695547392166094438411678515205767168001903165443774873634078727516192768
72097152587202565725224963053453312104962457602519098982444084232192575794053125667972710441984983040230917406729160359936248093081641104179231457280274877906944

All 15 rows

Technical details
Node id
n-062516b812f0ea15 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c94e3f620e523ccf

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,         32,          24;
  2:        64,        512,        1152,         1024,           320;
  3:       512,       6144,       23040,        40960,         38400,         18432,           3584;
  4:      4096,      65536,      344064,       917504,       1433600,       1376256,         802816,         262144,          36864;
  5:     32768,     655360,     4423680,     15728640,      34406400,      49545216,       48168960,       31457280,       13271040,        3276800,         360448;
  6:    262144,    6291456,    51904512,    230686720,     648806400,    1245708288,     1695547392,     1660944384,     1167851520,      576716800,      190316544,       37748736,        3407872;
  7:   2097152,   58720256,   572522496,   3053453312,   10496245760,   25190989824,    44084232192,    57579405312,    56679727104,    41984983040,    23091740672,     9160359936,     2480930816,      411041792,      31457280;
  8:  16777216,  536870912,  6039797760,  37580963840,  152672665600,  439697276928,   940463620096,  1535450808320,  1943304929280,  1919313510400,  1477871403008,   879394553856,   396948930560,   131533373440,   30198988800,    4294967296,    285212672;
  9: 134217728, 4831838208, 61605937152, 438086664192, 2053531238400, 6899864961024, 17441325318144, 34170759806976, 52857894076416, 65256659353600, 64604092760064, 51256139710464, 32391032733696, 16099684909056, 6160593715200, 1752346656768, 349100310528, 43486543872, 2550136832;
%F Formula
The row polynomials satisfy (1+(2*n-1)*x)*P(n,x) = (8+(16*n+24)*x+(32*n+24)*x^2+(16*n+8)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8 + (16*n + 24)*x + (32*n + 24)*x^2 + (16*n + 8)*x^3)*P[n - 1, x])/(1 + (2*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; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (16*n + 24)*x + (32*n + 24)*x^2 + (16*n + 8)*x^3)*v[i - 1])/(1 + (2*n - 1)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.