Real-rooted atlas

A131431 → x ↦ x+1 → D

palindromicγ-positivenonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2 - 5\,n + 3\,n^{2}\right)\,P_{n}(x) = \left(n + n x + 3\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2141428
3306030120
45215615652416
580320480320801280
6114570114011405701143648
71549242310308023109241549856
820014004200700070004200140020025600
9252201670561411217640141127056201625264512
1031027901116026040390603906026040111602790310158720
113743740168304488078540942487854044880168303740374382976
124444884244207326014652020512820512814652073260244204884444909312
135206240343201144002574004118404804804118402574001144003432062405202129920
14602782646956172172430430774774103303210330327747744304301721724695678266024931584

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2} - 9\,n^{2} - 18\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 10\,x - 24\,n - 24\,n x + 18\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)
Technical details
Node id
n-aa519c09f72bba51 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-3c03c6256e00c28a)
Fingerprint
f6-77a7ed3b1d5f257e

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:   4;
  2:  14,   14;
  3:  30,   60,   30;
  4:  52,  156,  156,    52;
  5:  80,  320,  480,   320,    80;
  6: 114,  570, 1140,  1140,   570,   114;
  7: 154,  924, 2310,  3080,  2310,   924,  154;
  8: 200, 1400, 4200,  7000,  7000,  4200, 1400,  200;
  9: 252, 2016, 7056, 14112, 17640, 14112, 7056, 2016, 252;
%F Formula
The row polynomials satisfy (3*n^2-5*n+2)*P(n,x) = (3*n^2+n+(3*n^2+n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 + n + (3*n^2 + n)*x)*P[n - 1, x])/(3*n^2 - 5*n + 2)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^2 + n + (3*n^2 + n)*x)*v[i - 1])/(3*n^2 - 5*n + 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.