Real-rooted atlas

A038327 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(169 + 52\,x + 4\,x^{2} + 26\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115217
2221604285
33211132618084725
4461372568853044801677625
5656903461370128440176801200321265625
69282325788283627682205137605304028806420503125
7130323843129952550551798521291836017981601485126720128330328125
8181970699320851814881039620400294292544516734405754112396032153602565296640625
9252876523513275472587418766633392623772240013243164481860243841726233610183683456051284575390625
103499484793095057530470203275472587401251108892803118861200052972657926200812804932096025459207680010241345517578125
1148250472147157698866544798556328351722024020132307606881098910401372298928001942330790419488268801356326406223360168960204821336064453125
12663099345793691158011331531609238639853757644506268137760144120793845603302927476992548919571200665941985285846480640361687040149360643686404096337340478515625
139086253197767591724058299063594150541473099108080068212065899228929074289544074942812799712143126857336322038844121600216431145216168898329609403863043530342479872081925319599853515625
1412417879370282373254415089537492522413681618689031614050537489250080560477484461294416200281602144643497331263986565725074293452871359544256006733413406724729153228823937105928237465617203201638483686387939453125
Technical details
Node id
n-2b7304c3ddb35f3e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d97455e8c3b8e2bb

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:          15,           2;
  2:         221,          60,           4;
  3:        3211,        1326,         180,          8;
  4:       46137,       25688,        5304,        480,         16;
  5:      656903,      461370,      128440,      17680,       1200,        32;
  6:     9282325,     7882836,     2768220,     513760,      53040,      2880,       64;
  7:   130323843,   129952550,    55179852,   12918360,    1798160,    148512,     6720,     128;
  8:  1819706993,  2085181488,  1039620400,  294292544,   51673440,   5754112,   396032,   15360,   256;
  9: 25287652351, 32754725874, 18766633392, 6237722400, 1324316448, 186024384, 17262336, 1018368, 34560, 512;
%F Formula
The row polynomials satisfy (2*n+11+2*x)*P(n,x) = (26*n+169+(4*n+52)*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[((26*n + 169 + (4*n + 52)*x + 4*x^2)*P[n - 1, x])/(2*n + 11 + 2*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] = ((26*n + 169 + (4*n + 52)*x + 4*x^2)*v[i - 1])/(2*n + 11 + 2*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

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.