Real-rooted atlas

A038208 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 3\,n x\right)\,P_{n}(x) = \left(8 + 24\,x + 18\,x^{2} + 12\,n x + 18\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141216
21696108220
36457612968642800
425630721036813824648034000
51024153606912013824012960046656400000
64096737284147201105920155520011197443265924600000
7163843440642322432774144014515200156764169144576223948852000000
865536157286412386304495452161161216001672151041463132167166361615116544580000000
92621447077888637009922972712968360755201504935936175575859212899450885441955841007769606400000000
101048576314572803185049601698693120557383680012039487488175575859201719926784010883911680403107840066512793670000000000
11419430413841203215571353609342812160350355456008828957491215450675609618919194624015963070464088683724800292656291844353564672760000000000
12167772166039797767474249728498283315202102132736006054142279681236054048768181624268390419155684556801418939596800702375100416208971104256282981703688200000000000
1367108864261724569635332816896259107323904121456558080039351924817929182115790848157407699271681992191193907218446214758400121745017405445433248710656147150485913618284971622488000000000000
142684354561127428915216488647884813190918307846801567252480244856421089286427481053593612592615941734418593784476467220659760529408017044302436761610142064259891241202136055808102395841085441175462461440940000000000000
Technical details
Node id
n-5e37a3501d591fe9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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:      4,      12;
  2:     16,      96,      108;
  3:     64,     576,     1296,       864;
  4:    256,    3072,    10368,     13824,      6480;
  5:   1024,   15360,    69120,    138240,    129600,      46656;
  6:   4096,   73728,   414720,   1105920,   1555200,    1119744,     326592;
  7:  16384,  344064,  2322432,   7741440,  14515200,   15676416,    9144576,    2239488;
  8:  65536, 1572864, 12386304,  49545216, 116121600,  167215104,  146313216,   71663616,  15116544;
  9: 262144, 7077888, 63700992, 297271296, 836075520, 1504935936, 1755758592, 1289945088, 544195584, 100776960;
%F Formula
The row polynomials satisfy (2+3*n*x)*P(n,x) = (8+(12*n+24)*x+(18*n+18)*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[((8 + (12*n + 24)*x + (18*n + 18)*x^2)*P[n - 1, x])/(2 + 3*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] = ((8 + (12*n + 24)*x + (18*n + 18)*x^2)*v[i - 1])/(2 + 3*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.