Real-rooted atlas

A249251 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-78 - 180\,x - 138\,x^{2} - 36\,x^{3} + 26\,n + 73\,n x + 53\,n x^{2} - 6\,n x^{3} - 8\,n x^{4} + 13\,n^{2} x + 42\,n^{2} x^{2} + 36\,n^{2} x^{3} + 8\,n^{2} x^{4} + 3\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n}(x) = \left(-78 - 180\,x - 138\,x^{2} - 36\,x^{3} - 130\,n - 404\,n x - 385\,n x^{2} - 108\,n x^{3} + 4\,n x^{4} - 104\,n^{2} x - 186\,n^{2} x^{2} - 84\,n^{2} x^{3} - 4\,n^{2} x^{4} - 15\,n^{3} x^{2} - 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(169 + 819\,x + 1558\,x^{2} + 1480\,x^{3} + 732\,x^{4} + 176\,x^{5} + 16\,x^{6} + 169\,n x + 624\,n x^{2} + 863\,n x^{3} + 548\,n x^{4} + 156\,n x^{5} + 16\,n x^{6} + 39\,n^{2} x^{2} + 87\,n^{2} x^{3} + 60\,n^{2} x^{4} + 12\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
213361867
34317417148436
414275611076241202749
5469309660215100198028816954
61549122042959233456195905760672102823
751164684213605319240815076566960157921536615472

All 15 rows

Technical details
Node id
n-5a205b819d0ad2e5 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-080cc94aee55e045

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:     4,      6;
  2:    13,     36,      18;
  3:    43,    174,     171,      48;
  4:   142,    756,    1107,     624,     120;
  5:   469,   3096,    6021,    5100,    1980,     288;
  6:  1549,  12204,   29592,   33456,   19590,    5760,     672;
  7:  5116,  46842,  136053,  192408,  150765,   66960,   15792,   1536;
  8: 16897, 176292,  596583, 1012944,  994725,  591984,  211008,  41472,   3456;
  9: 55807, 653538, 2524941, 5003532, 5907810, 4411170, 2108232, 626304, 105408, 7680;
%F Formula
The row polynomials satisfy (26*n-78+(13*n^2+73*n-180)*x+(3*n^3+42*n^2+53*n-138)*x^2+(6*n^3+36*n^2-6*n-36)*x^3+(8*n^2-8*n)*x^4)*P(n,x) = -(130*n+78+(104*n^2+404*n+180)*x+(15*n^3+186*n^2+385*n+138)*x^2+(12*n^3+84*n^2+108*n+36)*x^3+(4*n^2-4*n)*x^4)*P(n-1,x) + (169+(169*n+819)*x+(39*n^2+624*n+1558)*x^2+(87*n^2+863*n+1480)*x^3+(60*n^2+548*n+732)*x^4+(12*n^2+156*n+176)*x^5+(16*n+16)*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[(-(130*n + 78 + (104*n^2 + 404*n + 180)*x + (15*n^3 + 186*n^2 + 385*n + 138)*x^2 + (12*n^3 + 84*n^2 + 108*n + 36)*x^3 + (4*n^2 - 4*n)*x^4)*P[n - 1, x] + (169 + (169*n + 819)*x + (39*n^2 + 624*n + 1558)*x^2 + (87*n^2 + 863*n + 1480)*x^3 + (60*n^2 + 548*n + 732)*x^4 + (12*n^2 + 156*n + 176)*x^5 + (16*n + 16)*x^6)*D[P[n - 1, x], x])/(26*n - 78 + (13*n^2 + 73*n - 180)*x + (3*n^3 + 42*n^2 + 53*n - 138)*x^2 + (6*n^3 + 36*n^2 - 6*n - 36)*x^3 + (8*n^2 - 8*n)*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] = (-(130*n + 78 + (104*n^2 + 404*n + 180)*x + (15*n^3 + 186*n^2 + 385*n + 138)*x^2 + (12*n^3 + 84*n^2 + 108*n + 36)*x^3 + (4*n^2 - 4*n)*x^4)*v[i - 1] + (169 + (169*n + 819)*x + (39*n^2 + 624*n + 1558)*x^2 + (87*n^2 + 863*n + 1480)*x^3 + (60*n^2 + 548*n + 732)*x^4 + (12*n^2 + 156*n + 176)*x^5 + (16*n + 16)*x^6)*deriv(v[i - 1], x))/(26*n - 78 + (13*n^2 + 73*n - 180)*x + (3*n^3 + 42*n^2 + 53*n - 138)*x^2 + (6*n^3 + 36*n^2 - 6*n - 36)*x^3 + (8*n^2 - 8*n)*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.