Real-rooted atlas

A035607 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 18\,x - 46\,x^{2} - 16\,x^{3} + 32\,x^{4} + 2\,n x - 4\,n x^{2} - 40\,n x^{3} - 48\,n x^{4} + 8\,n^{2} x^{2} + 8\,n^{2} x^{3} + 16\,n^{2} x^{4} + 8\,n^{3} x^{3}\right)\,P_{n}(x) = \left(2 + 18\,x + 46\,x^{2} + 16\,x^{3} - 32\,x^{4} - 4\,n - 22\,n x - 24\,n x^{2} + 24\,n x^{3} + 48\,n x^{4} - 12\,n^{2} x - 56\,n^{2} x^{2} - 56\,n^{2} x^{3} - 16\,n^{2} x^{4} - 8\,n^{3} x^{2} - 24\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(1 + 14\,x + 74\,x^{2} + 180\,x^{3} + 192\,x^{4} + 64\,x^{5} + 4\,n x + 44\,n x^{2} + 176\,n x^{3} + 312\,n x^{4} + 240\,n x^{5} + 64\,n x^{6} + 4\,n^{2} x^{2} + 32\,n^{2} x^{3} + 72\,n^{2} x^{4} + 32\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
231619
376496167
4172246725121425
54170431685120256011593
69920961257631232332801228891571
723960164512015155225088019660857344707759
8577168321514886420481448960176947210895362621445381057
91393462084851842484224711680011894784114114565767168117964840386865
1033631250081500096900608031313920666746888767897668943872294912005242880299980083
118119334272451267231084544127196160329711616553426944596639744396361728146800640230686722209145111
1219601885408132811201032432644861593601486946304304433561641929932803817340928219152384071512883210066329616152520849
13473212326848383929923324712961771153920624771072015119835136254295408642958439219223257415680117419540483422552064436207616117384000697
14114243607448010934064010437811206207040000248174714886937907200013808828416019628752896019715850240013617437081661303947264161396817921879048192848594257555
Technical details
Node id
n-96216ae51b129dec · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-04a28eeb4921500f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
  2:    3,    16;
  3:    7,    64,     96;
  4:   17,   224,    672,     512;
  5:   41,   704,   3168,    5120,    2560;
  6:   99,  2096,  12576,   31232,   33280,    12288;
  7:  239,  6016,  45120,  151552,  250880,   196608,    57344;
  8:  577, 16832, 151488,  642048, 1448960,  1769472,  1089536,  262144;
  9: 1393, 46208, 485184, 2484224, 7116800, 11894784, 11411456, 5767168, 1179648;
%F Formula
The row polynomials satisfy -(2-(2*n-18)*x-(8*n^2-4*n-46)*x^2-(8*n^3+8*n^2-40*n-16)*x^3-(16*n^2-48*n+32)*x^4)*P(n,x) = -(4*n-2+(12*n^2+22*n-18)*x+(8*n^3+56*n^2+24*n-46)*x^2+(24*n^3+56*n^2-24*n-16)*x^3+(16*n^2-48*n+32)*x^4)*P(n-1,x) + (1+(4*n+14)*x+(4*n^2+44*n+74)*x^2+(32*n^2+176*n+180)*x^3+(72*n^2+312*n+192)*x^4+(32*n^2+240*n+64)*x^5+64*n*x^6)*d/dx 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[(-(4*n - 2 + (12*n^2 + 22*n - 18)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (24*n^3 + 56*n^2 - 24*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)*P[n - 1, x] + (1 + (4*n + 14)*x + (4*n^2 + 44*n + 74)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (72*n^2 + 312*n + 192)*x^4 + (32*n^2 + 240*n + 64)*x^5 + 64*n*x^6)*D[P[n - 1, x], x])/(-2 + (2*n - 18)*x + (8*n^2 - 4*n - 46)*x^2 + (8*n^3 + 8*n^2 - 40*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)]];
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] = (-(4*n - 2 + (12*n^2 + 22*n - 18)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (24*n^3 + 56*n^2 - 24*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)*v[i - 1] + (1 + (4*n + 14)*x + (4*n^2 + 44*n + 74)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (72*n^2 + 312*n + 192)*x^4 + (32*n^2 + 240*n + 64)*x^5 + 64*n*x^6)*deriv(v[i - 1], x))/(-2 + (2*n - 18)*x + (8*n^2 - 4*n - 46)*x^2 + (8*n^3 + 8*n^2 - 40*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)); 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

References

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