Real-rooted atlas

A130123 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
188
232128160
39676815362400
42563072122881638432000
56401024061440163840163840400000
6153630720245760983040196608015728644800000
7358486016860160458752013762560220200961468006456000000
8819222937627525121835008073400320176160768234881024134217728640000000
91843258982482575366606028833030144010569646082113929216241591910412079595527200000000
1040960147456023592960220200960132120576052848230401409286144024159191040241591910401073741824080000000000
11901123604480648806406920601604844421120232532213767751073792017716740096026575110144023622320128094489280512880000000000
1219660886507521730150402076180480166094438409301288550437205154201610630044057602126008811520283467841536022677427322888246337208329600000000000
134259842044723244983910459978547205398069248034547643187216122233487365527622909952138190572748802456721293312029480655519744214404767416327146825580544104000000000000
1491750447710208114504499216793993216167939932160120916751155264488933949442579557357977677386720739328171970490531840275152784850944300166674382848200111116255232615726511554561120000000000000
Technical details
Node id
n-bcadabd274f97937 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-03d7b8417365c509

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:     0;
  1:     8;
  2:    32,    128;
  3:    96,    768,    1536;
  4:   256,   3072,   12288,    16384;
  5:   640,  10240,   61440,   163840,    163840;
  6:  1536,  30720,  245760,   983040,   1966080,    1572864;
  7:  3584,  86016,  860160,  4587520,  13762560,   22020096,   14680064;
  8:  8192, 229376, 2752512, 18350080,  73400320,  176160768,  234881024,  134217728;
  9: 18432, 589824, 8257536, 66060288, 330301440, 1056964608, 2113929216, 2415919104, 1207959552;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+8*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 8*n*x)*v[i - 1])/(n - 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. [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.