Real-rooted atlas

A027466 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + 3\,n x\right)\,P_{n}(x) = \left(64 + 48\,x + 9\,x^{2} + 24\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18614
2649627187
351211526481082420
440961228810368345640530613
53276812288013824069120162001458380666
62621441179648165888011059203888006998451034670479
72097152110100481857945615482880725760019595522857681749656689952
816777216100663296198180864198180864116121600418037769144576111974459049682050985
91342177289059696642038431744237817036816721510407524679682194698244031078442515281968308145637478
101073741824805306368020384317440271790899202229534720012039487488438939648010749542401700611201574640064953996675855331
11858993459270866960384199313326080298969989120280284364800176579149824772533780482364899328049884595206928416005715943221257641141246682444
126871947673661847529062419134079303683189013217280336341237760024216569118721236054048768454060670976119723028480221709312002743652736204073344690873313409648518717
135497558138885360119185408180904022507523316573745971238866098585600314815398543361836423158169678703849635842490238992384576444211200951132948481061181388871850823222320522156921418836050
1443980465111044617948836659216884375434035233768750868070443530030415872039177027374284825709924214374412592615941734446484461191168129123503308802663172255744396174385152402364609922499898464717445351829703743628343
Technical details
Node id
n-aad4518e3f6b8898 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-969401a978cdcdca

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,         6;
  2:        64,        96,         27;
  3:       512,      1152,        648,        108;
  4:      4096,     12288,      10368,       3456,        405;
  5:     32768,    122880,     138240,      69120,      16200,      1458;
  6:    262144,   1179648,    1658880,    1105920,     388800,     69984,      5103;
  7:   2097152,  11010048,   18579456,   15482880,    7257600,   1959552,    285768,    17496;
  8:  16777216, 100663296,  198180864,  198180864,  116121600,  41803776,   9144576,  1119744,   59049;
  9: 134217728, 905969664, 2038431744, 2378170368, 1672151040, 752467968, 219469824, 40310784, 4251528, 196830;
%F Formula
The row polynomials satisfy (8+3*n*x)*P(n,x) = (64+(24*n+48)*x+(9*n+9)*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[((64 + (24*n + 48)*x + (9*n + 9)*x^2)*P[n - 1, x])/(8 + 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] = ((64 + (24*n + 48)*x + (9*n + 9)*x^2)*v[i - 1])/(8 + 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.