Real-rooted atlas

A038327 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + x + n\right)\,P_{n}(x) = \left(144 + 24\,x + x^{2} + 12\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113114
2168261195
321605043912704
4276488640100852137349
5352512138240216001680651514098
6447897621150724147204320025207817054567
75673369631352832740275296768075600352891196536180
8716636160453869568125411328197406721935360120960470410411317718857
99029615616644972544020424130563762339844441651234836481814406048117117946075862
101135151677449029615616032248627200680804352094058496088833024580608025920075601301243903485579
11142409937715212486668451844966288588801182449664001872211968020692869121628605449123840356400924014313308603804376
121783220089651217089192525824749200107110419865154355203547348992004493308723241385738242791895041368576047520011088156144804009850925
1322290251120640023181861165465611107975141785632465337974784645617516544092231073792097355022336768592281645368294419768320617760131041691605750213184506
1427818233398558723120635156889600162273028158259251837217328332811362868291174418077290463232215205838848019471004467213450364928705729024276756487862401528818218177627877990831
Technical details
Node id
n-9c65166b4e60b235 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ba08af6d0860fb7c

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:         13,          1;
  2:        168,         26,          1;
  3:       2160,        504,         39,         1;
  4:      27648,       8640,       1008,        52,        1;
  5:     352512,     138240,      21600,      1680,       65,       1;
  6:    4478976,    2115072,     414720,     43200,     2520,      78,      1;
  7:   56733696,   31352832,    7402752,    967680,    75600,    3528,     91,    1;
  8:  716636160,  453869568,  125411328,  19740672,  1935360,  120960,   4704,  104,   1;
  9: 9029615616, 6449725440, 2042413056, 376233984, 44416512, 3483648, 181440, 6048, 117, 1;
%F Formula
The row polynomials satisfy (n+11+x)*P(n,x) = (12*n+144+(n+24)*x+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[((12*n + 144 + (n + 24)*x + x^2)*P[n - 1, x])/(n + 11 + 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] = ((12*n + 144 + (n + 24)*x + x^2)*v[i - 1])/(n + 11 + 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.