Real-rooted atlas

A038259 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + 5\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
155
211050160
3181516503753840
4266203630016500250081920
5366025665500453750137500156251638400
64831530109807509982500453750010312509375031457280
762004635169103550192163125116462500397031257218750546875587202560
8779486840248018540033820710002562175000116462500031762500048125000312500010737418240
996461496453507690780055804171500507310650002882446875010481625000238218750030937500017578125193273528320
1011789738455048230748225087692269500093006952500063413831250028824468750087346875000170156250001933593750976562503435973836800
1114265583530556484356150250132634557618751607691607500012788455968750697552143750026422429687506862968750001169824218751181640625053710937560473139527680
121711870023666085593501183300194530684507500265269115237500241153741125000153461471625000697552143750002264779687500051472265625007798828125007089843750029296875001055531162664960
132039978444868651112715515382900278178878845725042148314976625004310623122609375313499863462500016624992759375006477269906250001840133496093753717441406250050692382812504189453125001586914062518295873486192640
142416589850075170142798491140805503894504303840150064908405064002500737595512090937506034872371653125036574984070625000166249927593750005667611167968750143121494140625026022089843750032258789062500244384765625085449218750315251973915934720
Technical details
Node id
n-132cb39dd7c496a1 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2235b7dd30e0ad6a

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:          5;
  2:        110,          50;
  3:       1815,        1650,         375;
  4:      26620,       36300,       16500,        2500;
  5:     366025,      665500,      453750,      137500,       15625;
  6:    4831530,    10980750,     9982500,     4537500,     1031250,       93750;
  7:   62004635,   169103550,   192163125,   116462500,    39703125,     7218750,     546875;
  8:  779486840,  2480185400,  3382071000,  2562175000,  1164625000,   317625000,   48125000,   3125000;
  9: 9646149645, 35076907800, 55804171500, 50731065000, 28824468750, 10481625000, 2382187500, 309375000, 17578125;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+5*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 5*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] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 5*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.