Real-rooted atlas

A139548 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12432864
232010241152512643072
33584184323840040960230406144512131072
436864262144802816137625614336009175043440646553640965242880
53604483276800132710403145728048168960495452163440640015728640442368065536032768201326592
634078723774873619031654457671680011678515201660944384169554739212457082886488064002306867205190451262914562621447516192768
73145728041104179224809308169160359936230917406724198498304056679727104575794053124408423219225190989824104962457603053453312572522496587202562097152274877906944

All 15 rows

Technical details
Node id
n-858b0d55747d5753 · revision 0 · JSON
Definition source
derived:one_plus_derivative
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:         24,          32,            8;
  2:        320,        1024,         1152,           512,            64;
  3:       3584,       18432,        38400,         40960,         23040,           6144,            512;
  4:      36864,      262144,       802816,       1376256,       1433600,         917504,         344064,          65536,           4096;
  5:     360448,     3276800,     13271040,      31457280,      48168960,       49545216,       34406400,       15728640,        4423680,         655360,          32768;
  6:    3407872,    37748736,    190316544,     576716800,    1167851520,     1660944384,     1695547392,     1245708288,      648806400,      230686720,       51904512,        6291456,         262144;
  7:   31457280,   411041792,   2480930816,    9160359936,   23091740672,    41984983040,    56679727104,    57579405312,    44084232192,    25190989824,    10496245760,     3053453312,      572522496,      58720256,       2097152;
  8:  285212672,  4294967296,  30198988800,  131533373440,  396948930560,   879394553856,  1477871403008,  1919313510400,  1943304929280,  1535450808320,   940463620096,   439697276928,   152672665600,   37580963840,    6039797760,    536870912,    16777216;
  9: 2550136832, 43486543872, 349100310528, 1752346656768, 6160593715200, 16099684909056, 32391032733696, 51256139710464, 64604092760064, 65256659353600, 52857894076416, 34170759806976, 17441325318144, 6899864961024, 2053531238400, 438086664192, 61605937152, 4831838208, 134217728;
%F Formula
The row polynomials satisfy (2*n-1+x)*P(n,x) = (16*n+8+(32*n+24)*x+(16*n+24)*x^2+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[((16*n + 8 + (32*n + 24)*x + (16*n + 24)*x^2 + 8*x^3)*P[n - 1, x])/(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] = ((16*n + 8 + (32*n + 24)*x + (16*n + 24)*x^2 + 8*x^3)*v[i - 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.