Real-rooted atlas

image under θ+1 → θ+1 → D → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n^{2} - 2\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-n^{2} - 3\,n^{2} x - n^{2} x^{2} - 2\,n^{3} - 6\,n^{3} x - 2\,n^{3} x^{2} - n^{4} - 3\,n^{4} x - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-3\,n^{2} x - 2\,n^{2} x^{2} - 6\,n^{3} x - 4\,n^{3} x^{2} - 3\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-2}'(x) + \left(-n^{2} x^{2} - 2\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-2}''(x) + \left(2 + 2\,x - 4\,n^{2} - 4\,n^{2} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(2\,x - 4\,n^{2} x + 2\,n^{4} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2181836
34814448240
41007006001001500
51802700450018001809360
629491142646019110441029459682
744828224134848156800627209408448392896
8648822966259681102248680400172368181446482682720
990022950027225006993000625590023814004158003240090019032300
101210618310112893004126705051455250276206707114800907500544501210140329750
1115841620432451455842308680003908203202843074081013554081881792018295208712015841074854880
1220284151316175474728124012402827974232002684366256127235908832250675645173700345774013384820288545174716
13254810434060666620500645288644019131660548237507029761456290035248402113769160110961002001006194188198744254870438025476
1431502580165024852555003273399675012623636025019982457495015538723200065873808000161756595002371168800208107900106060502866503150601332864300
Technical details
Node id
n-a0ffac8277c87049 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-4732d0911d5a78c5

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: 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:   0;
  1:   4;
  2:  18,     18;
  3:  48,    144,      48;
  4: 100,    700,     600,     100;
  5: 180,   2700,    4500,    1800,     180;
  6: 294,   9114,   26460,   19110,    4410,     294;
  7: 448,  28224,  134848,  156800,   62720,    9408,    448;
  8: 648,  82296,  625968, 1102248,  680400,  172368,  18144,   648;
  9: 900, 229500, 2722500, 6993000, 6255900, 2381400, 415800, 32400, 900;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n^2)*P(n,x) = -(n^4+2*n^3+n^2+(3*n^4+6*n^3+3*n^2)*x+(n^4+2*n^3+n^2)*x^2)*P(n-2,x) - ((3*n^4+6*n^3+3*n^2)*x+(2*n^4+4*n^3+2*n^2)*x^2)*d/dx P(n-2,x) - (n^4+2*n^3+n^2)*x^2*d^2/dx^2 P(n-2,x) + (2*n^4-4*n^2+2+(2*n^4-4*n^2+2)*x)*P(n-1,x) + (2*n^4-4*n^2+2)*x*d/dx 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[(-(n^4 + 2*n^3 + n^2 + (3*n^4 + 6*n^3 + 3*n^2)*x + (n^4 + 2*n^3 + n^2)*x^2)*P[n - 2, x] - ((3*n^4 + 6*n^3 + 3*n^2)*x + (2*n^4 + 4*n^3 + 2*n^2)*x^2)*D[P[n - 2, x], x] - (n^4 + 2*n^3 + n^2)*x^2*D[P[n - 2, x], {x, 2}] + (2*n^4 - 4*n^2 + 2 + (2*n^4 - 4*n^2 + 2)*x)*P[n - 1, x] + (2*n^4 - 4*n^2 + 2)*x*D[P[n - 1, x], x])/(n^4 - 2*n^3 + 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] = (-(n^4 + 2*n^3 + n^2 + (3*n^4 + 6*n^3 + 3*n^2)*x + (n^4 + 2*n^3 + n^2)*x^2)*v[i - 2] - ((3*n^4 + 6*n^3 + 3*n^2)*x + (2*n^4 + 4*n^3 + 2*n^2)*x^2)*deriv(v[i - 2], x) - (n^4 + 2*n^3 + n^2)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n^4 - 4*n^2 + 2 + (2*n^4 - 4*n^2 + 2)*x)*v[i - 1] + (2*n^4 - 4*n^2 + 2)*x*deriv(v[i - 1], x))/(n^4 - 2*n^3 + 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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.