Real-rooted atlas

A027467 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 - 8\,x + 24\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(1024 + 384\,x + 48\,x^{2} + 2\,x^{3} + 384\,n x + 80\,n x^{2} + 4\,n x^{3} + 16\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
116824
225625636548
340966144172812812096
465536131072552968192400260496
51048576262144014745603276803200011525505408
61677721650331648353894401048576015360001105923136114633792
726843545693952409679272345629360128057344000619315235123281922358180864

All 15 rows

Technical details
Node id
n-90fcc7328f98febf · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5484980da109477c

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:          16,            8;
  2:         256,          256,           36;
  3:        4096,         6144,         1728,          128;
  4:       65536,       131072,        55296,         8192,         400;
  5:     1048576,      2621440,      1474560,       327680,       32000,       1152;
  6:    16777216,     50331648,     35389440,     10485760,     1536000,     110592,       3136;
  7:   268435456,    939524096,    792723456,    293601280,    57344000,    6193152,     351232,     8192;
  8:  4294967296,  17179869184,  16911433728,   7516192768,  1835008000,  264241152,   22478848,  1048576,   20736;
  9: 68719476736, 309237645312, 347892350976, 180388626432, 52848230400, 9512681472, 1078984704, 75497472, 2985984, 51200;
%F Formula
The row polynomials satisfy (64+(24*n-8)*x+n^2*x^2)*P(n,x) = (1024+(384*n+384)*x+(16*n^2+80*n+48)*x^2+(2*n^2+4*n+2)*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[((1024 + (384*n + 384)*x + (16*n^2 + 80*n + 48)*x^2 + (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x])/(64 + (24*n - 8)*x + n^2*x^2)]];
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] = ((1024 + (384*n + 384)*x + (16*n^2 + 80*n + 48)*x^2 + (2*n^2 + 4*n + 2)*x^3)*v[i - 1])/(64 + (24*n - 8)*x + n^2*x^2)); 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.