Real-rooted atlas

A131431 → x ↦ x+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(2 + 3\,x + 6\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181220
2288463175
3803605402701250
420812482808280810538125
55123840115201728012960388850000
61216109444104082080923405540413851296875
7281629568133056332640498960449064224532481141718750
864007680040320012096002268000272160020412008748001640259765625
91433619353611612164064256914457613716864137168648817984330674455112454687500
10317444761603214080128563203374784060746112759326406508512036610380122034601830519302734375
116963211489288616960387763201163289602442908163664362243926102402944576801472288404416865260229981660156250
121515522727936225054721125273603797798409114716161595075328205081113619226354401281756960576790632157306536196633179033203125
133276806389760575078403162931201186099200320246784064049356809607403520108083289609006940800540416448022107945605526986406377292048828125000
147045121479475214424883286549299235701585921071047577624098570496413118351365422178361654221783616406663377122218163875283181145321919564892205667667262451171875
Technical details
Node id
n-e06e654167609cff · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f06e51c139cfae05

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:     8,     12;
  2:    28,     84,      63;
  3:    80,    360,     540,     270;
  4:   208,   1248,    2808,    2808,    1053;
  5:   512,   3840,   11520,   17280,   12960,     3888;
  6:  1216,  10944,   41040,   82080,   92340,    55404,    13851;
  7:  2816,  29568,  133056,  332640,  498960,   449064,   224532,   48114;
  8:  6400,  76800,  403200, 1209600, 2268000,  2721600,  2041200,  874800,  164025;
  9: 14336, 193536, 1161216, 4064256, 9144576, 13716864, 13716864, 8817984, 3306744, 551124;
%F Formula
The row polynomials satisfy (3*n-2)*P(n,x) = (6*n+2+(9*n+3)*x)*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[((6*n + 2 + (9*n + 3)*x)*P[n - 1, x])/(3*n - 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] = ((6*n + 2 + (9*n + 3)*x)*v[i - 1])/(3*n - 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.