Real-rooted atlas

A131431 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + 3\,n - 9\,n x + 9\,n^{2} x - 2\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(1 + 3\,x + 3\,x^{2} + x^{3} + 3\,n + 12\,n x + 14\,n x^{2} + 5\,n x^{3} + 9\,n^{2} x + 16\,n^{2} x^{2} + 7\,n^{2} x^{3} + 3\,n^{3} x^{2} + 3\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141620
275663126
310120270160560
4132087028323252080
5163201440256020005766912
619456256560807125410493121280
72261641581232019250166327546140861952
825800630022400437505040034300128002025172800
92810089072376328820012700811524864512204122800465920
1031124012555595201627502812323189902380801129953100037511222144
1134149616830897602805005654887696927180804544101870004525448963133440
123717762197813024045787510549441675212187545614835158140002954826393662537880704
134020802808018304071500018532803363360439296041698802860000138424044928087880784019496960
1443240835217250432107607530990966327321944486410459449860860052082032253888661297117992967547554560
Technical details
Node id
n-f22362a5eb086e6f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  4,   16;
  2:  7,   56,   63;
  3: 10,  120,  270,   160;
  4: 13,  208,  702,   832,   325;
  5: 16,  320, 1440,  2560,  2000,    576;
  6: 19,  456, 2565,  6080,  7125,   4104,    931;
  7: 22,  616, 4158, 12320, 19250,  16632,   7546,  1408;
  8: 25,  800, 6300, 22400, 43750,  50400,  34300, 12800,  2025;
  9: 28, 1008, 9072, 37632, 88200, 127008, 115248, 64512, 20412, 2800;
%F Formula
The row polynomials satisfy (3*n-2+(9*n^2-9*n+2)*x+(3*n^3-2*n^2)*x^2)*P(n,x) = (3*n+1+(9*n^2+12*n+3)*x+(3*n^3+16*n^2+14*n+3)*x^2+(3*n^3+7*n^2+5*n+1)*x^3)*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^2 + 12*n + 3)*x + (3*n^3 + 16*n^2 + 14*n + 3)*x^2 + (3*n^3 + 7*n^2 + 5*n + 1)*x^3)*P[n - 1, x])/(3*n - 2 + (9*n^2 - 9*n + 2)*x + (3*n^3 - 2*n^2)*x^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] = ((3*n + 1 + (9*n^2 + 12*n + 3)*x + (3*n^3 + 16*n^2 + 14*n + 3)*x^2 + (3*n^3 + 7*n^2 + 5*n + 1)*x^3)*v[i - 1])/(3*n - 2 + (9*n^2 - 9*n + 2)*x + (3*n^3 - 2*n^2)*x^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.