Real-rooted atlas

A127773 → D → θ+1

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(36 - 132\,n + 193\,n^{2} - 144\,n^{3} + 58\,n^{4} - 12\,n^{5} + n^{6}\right)\,P_{n}(x) = \left(4\,n^{2} x^{3} - 12\,n^{3} x^{3} + 13\,n^{4} x^{3} - 6\,n^{5} x^{3} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(-27\,n^{2} x^{2} + 72\,n^{3} x^{2} - 66\,n^{4} x^{2} + 24\,n^{5} x^{2} - 3\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(108\,n^{2} x - 180\,n^{3} x + 111\,n^{4} x - 30\,n^{5} x + 3\,n^{6} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
\(P_{2}(x) = 24\,x\)
\(P_{3}(x) = 90\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
202424
3009090
4000240240
50000525525
60000010081008
700000017641764
8000000028802880
90000000044554455
1000000000066006600
11000000000094389438
12000000000001310413104
130000000000001774517745
1400000000000002352023520
Technical details
Node id
n-19c0690cfb4c0a03 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-edf6954a4cd8adfc

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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: 0;
  1: 3;
  2: 0, 24;
  3: 0,  0, 90;
  4: 0,  0,  0, 240;
  5: 0,  0,  0,   0, 525;
  6: 0,  0,  0,   0,   0, 1008;
  7: 0,  0,  0,   0,   0,    0, 1764;
  8: 0,  0,  0,   0,   0,    0,    0, 2880;
  9: 0,  0,  0,   0,   0,    0,    0,    0, 4455;
%F Formula
The row polynomials satisfy (n^6-12*n^5+58*n^4-144*n^3+193*n^2-132*n+36)*P(n,x) = (n^6-6*n^5+13*n^4-12*n^3+4*n^2)*x^3*P(n-3,x) - (3*n^6-24*n^5+66*n^4-72*n^3+27*n^2)*x^2*P(n-2,x) + (3*n^6-30*n^5+111*n^4-180*n^3+108*n^2)*x*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 24*x, P(3,x) = 90*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 24*x;
P[3, x_] := 90*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^3*P[n - 3, x] - (3*n^6 - 24*n^5 + 66*n^4 - 72*n^3 + 27*n^2)*x^2*P[n - 2, x] + (3*n^6 - 30*n^5 + 111*n^4 - 180*n^3 + 108*n^2)*x*P[n - 1, x])/(n^6 - 12*n^5 + 58*n^4 - 144*n^3 + 193*n^2 - 132*n + 36)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 24*x; v[4] = 90*x^2; for(n = 4, N, my(i = n + 1); v[i] = ((n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^3*v[i - 3] - (3*n^6 - 24*n^5 + 66*n^4 - 72*n^3 + 27*n^2)*x^2*v[i - 2] + (3*n^6 - 30*n^5 + 111*n^4 - 180*n^3 + 108*n^2)*x*v[i - 1])/(n^6 - 12*n^5 + 58*n^4 - 144*n^3 + 193*n^2 - 132*n + 36)); 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.