Real-rooted atlas

A081580 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n x + 24\,n x^{2} + 16\,n x^{3} + 12\,n^{2} x^{2} + 24\,n^{2} x^{3} + 8\,n^{3} x^{3}\right)\,P_{n}(x) = \left(12 + 114\,x + 344\,x^{2} + 352\,x^{3} + 128\,x^{4} - 12\,n - 66\,n x - 20\,n x^{2} + 264\,n x^{3} + 192\,n x^{4} - 36\,n^{2} x - 180\,n^{2} x^{2} - 144\,n^{2} x^{3} + 64\,n^{2} x^{4} - 24\,n^{3} x^{2} - 88\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3 + 42\,x + 218\,x^{2} + 500\,x^{3} + 448\,x^{4} + 64\,x^{5} + 12\,n x + 132\,n x^{2} + 488\,n x^{3} + 632\,n x^{4} + 96\,n x^{5} + 12\,n^{2} x^{2} + 96\,n^{2} x^{3} + 200\,n^{2} x^{4} + 32\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
297248129
3283367682561388
4101161659525888128014837
534268403552065280384006144152526
6118928536190320508160556800227328286721541005
74088114464934656330496054784004079616126156813107215308824

All 15 rows

Technical details
Node id
n-361908edb9a57e8f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-fa76549ffa31a198

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:     2,       8;
  2:     9,      72,       48;
  3:    28,     336,      768,       256;
  4:   101,    1616,     5952,      5888,      1280;
  5:   342,    6840,    35520,     65280,     38400,      6144;
  6:  1189,   28536,   190320,    508160,    556800,    227328,     28672;
  7:  4088,  114464,   934656,   3304960,   5478400,   4079616,   1261568,    131072;
  8: 14121,  451872,  4363776,  19210752,  42969600,  48955392,  27009024,   6684672,   589824;
  9: 48682, 1752552, 19545984, 103143936, 292008960, 456044544, 383975424, 166330368, 34209792, 2621440;
%F Formula
The row polynomials satisfy (6*n*x+(12*n^2+24*n)*x^2+(8*n^3+24*n^2+16*n)*x^3)*P(n,x) = -(12*n-12+(36*n^2+66*n-114)*x+(24*n^3+180*n^2+20*n-344)*x^2+(88*n^3+144*n^2-264*n-352)*x^3-(64*n^2+192*n+128)*x^4)*P(n-1,x) + (3+(12*n+42)*x+(12*n^2+132*n+218)*x^2+(96*n^2+488*n+500)*x^3+(200*n^2+632*n+448)*x^4+(32*n^2+96*n+64)*x^5)*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[(-(12*n - 12 + (36*n^2 + 66*n - 114)*x + (24*n^3 + 180*n^2 + 20*n - 344)*x^2 + (88*n^3 + 144*n^2 - 264*n - 352)*x^3 - (64*n^2 + 192*n + 128)*x^4)*P[n - 1, x] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 218)*x^2 + (96*n^2 + 488*n + 500)*x^3 + (200*n^2 + 632*n + 448)*x^4 + (32*n^2 + 96*n + 64)*x^5)*D[P[n - 1, x], x])/(6*n*x + (12*n^2 + 24*n)*x^2 + (8*n^3 + 24*n^2 + 16*n)*x^3)]];
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] = (-(12*n - 12 + (36*n^2 + 66*n - 114)*x + (24*n^3 + 180*n^2 + 20*n - 344)*x^2 + (88*n^3 + 144*n^2 - 264*n - 352)*x^3 - (64*n^2 + 192*n + 128)*x^4)*v[i - 1] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 218)*x^2 + (96*n^2 + 488*n + 500)*x^3 + (200*n^2 + 632*n + 448)*x^4 + (32*n^2 + 96*n + 64)*x^5)*deriv(v[i - 1], x))/(6*n*x + (12*n^2 + 24*n)*x^2 + (8*n^3 + 24*n^2 + 16*n)*x^3)); 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.