Real-rooted atlas

A395972 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3} + 4\,n x + 10\,n x^{2} + 8\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 6\,x^{3} - 6\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 4 + 24\,x + 28\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
14242856
2896312272688
31632020164672334410368
4329609920435208016049472184064
564268841280293120100320015555848529283748864
612871681532161603840878528024933888337191681675801685960704
7256184325232647583744608025602735452166686955528076349443690826242187886592
851246080167731232169984355396608232539955289359872001930523443221180916224899692236861133750272

All 15 rows

Technical details
Node id
n-18383b729ceef751 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ced16c5d17801fb2

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    4,     24,       28;
   2:    8,     96,      312,       272;
   3:   16,    320,     2016,      4672,       3344;
   4:   32,    960,     9920,     43520,      80160,        49472;
   5:   64,   2688,    41280,    293120,    1003200,      1555584,       852928;
   6:  128,   7168,   153216,   1603840,    8785280,     24933888,     33719168,      16758016;
   7:  256,  18432,   523264,   7583744,   60802560,    273545216,    668695552,     807634944,      369082624;
   8:  512,  46080,  1677312,  32169984,  355396608,   2325399552,   8935987200,   19305234432,    21180916224,     8996922368;
   9: 1024, 112640,  5114880, 125460480, 1828055040,  16409530368,  91187927040,  307421306880,   597913850880,   603556771840,   240294124544;
  10: 2048, 270336, 14981120, 457543680, 8505446400, 100385488896, 762236547072, 3696951582720, 11147970324480, 19793229209600, 18564719720448, 6974172532736;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+(4*n+8)*x+(10*n+10)*x^2+(8*n+4)*x^3+2*n*x^4)*P(n-1,x) - (2*x^2+6*x^3+6*x^4+2*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 24*x + 28*x^2.
%t Mathematica
P[1, x_] := 4 + 24*x + 28*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (4*n + 8)*x + (10*n + 10)*x^2 + (8*n + 4)*x^3 + 2*n*x^4)*P[n - 1, x] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 4 + 24*x + 28*x^2; for(n = 2, N, my(i = n); v[i] = ((2 + (4*n + 8)*x + (10*n + 10)*x^2 + (8*n + 4)*x^3 + 2*n*x^4)*v[i - 1] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.