Real-rooted atlas

A131431 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
27282863
3106012080270
4131043124162081053
516160640128012805123888
6192281140304045603648121613851
7223081848616012320147849856281648114
825400280011200280004480044800256006400164025
928504403218816564481128961505281290246451214336551124
1031620558029760104160249984416640476160357120158720317441830519
1134748748044880179520502656100531214361601436160957440382976696326022998
12378889768651202930409377282188032375091246886404167680250060890931215155219663317
1340104012480915204576001647360439296087859201317888014643200117145606389760212992032768063772920
1443120415652125216688688275475282642561888972833057024440760324407603232055296160276484931584704512205667667
Technical details
Node id
n-0b91678f1ee08b15 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-11aec07ce60b6d61

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:  4,   8;
  2:  7,  28,   28;
  3: 10,  60,  120,    80;
  4: 13, 104,  312,   416,   208;
  5: 16, 160,  640,  1280,  1280,    512;
  6: 19, 228, 1140,  3040,  4560,   3648,   1216;
  7: 22, 308, 1848,  6160, 12320,  14784,   9856,   2816;
  8: 25, 400, 2800, 11200, 28000,  44800,  44800,  25600,  6400;
  9: 28, 504, 4032, 18816, 56448, 112896, 150528, 129024, 64512, 14336;
%F Formula
The row polynomials satisfy (3*n-2+(3*n-2)*x)*P(n,x) = (3*n+1+(9*n+3)*x+(6*n+2)*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[((3*n + 1 + (9*n + 3)*x + (6*n + 2)*x^2)*P[n - 1, x])/(3*n - 2 + (3*n - 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] = ((3*n + 1 + (9*n + 3)*x + (6*n + 2)*x^2)*v[i - 1])/(3*n - 2 + (3*n - 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

Candidate equalities ?

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.