Real-rooted atlas

A027465 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(x + n\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} + 4\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2486416128
3256576384641280
4128040964608204825612288
56144256004096030720102401024114688
6286721474563072003276801843204915240961048576
71310728028162064384286720022937601032192229376163849437184

All 15 rows

Technical details
Node id
n-76c02e9a4178807b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e99c3824c4864868

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       8,        4;
  2:      48,       64,       16;
  3:     256,      576,      384,        64;
  4:    1280,     4096,     4608,      2048,       256;
  5:    6144,    25600,    40960,     30720,     10240,      1024;
  6:   28672,   147456,   307200,    327680,    184320,     49152,     4096;
  7:  131072,   802816,  2064384,   2867200,   2293760,   1032192,   229376,    16384;
  8:  589824,  4194304, 12845056,  22020096,  22937600,  14680064,  5505024,  1048576,   65536;
  9: 2621440, 21233664, 75497472, 154140672, 198180864, 165150720, 88080384, 28311552, 4718592, 262144;
%F Formula
The row polynomials satisfy (n+x)*P(n,x) = (4*n+4+(4*n+8)*x+4*x^2)*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[((4*n + 4 + (4*n + 8)*x + 4*x^2)*P[n - 1, x])/(n + 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] = ((4*n + 4 + (4*n + 8)*x + 4*x^2)*v[i - 1])/(n + 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.