Real-rooted atlas

A128813 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 16\,n x - n^{2} - 8\,n^{2} x + 36\,n^{2} x^{2} - 3\,n^{3} - 8\,n^{3} x - 36\,n^{3} x^{2} + n^{4} + 8\,n^{4} x - 36\,n^{4} x^{2} - n^{5} - 8\,n^{5} x + 36\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-16\,n x^{2} + 8\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{3} x^{2} - 16\,n^{3} x^{3} - 8\,n^{4} x^{2} - 16\,n^{4} x^{3} + 8\,n^{5} x^{2} + 16\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(4\,n - 16\,n x + 4\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(16\,n x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
2880108196
356944291623046220
4616148647455612902472000291060
59856311744219621660407047200000311040018868920
62168328514176777765602968796165548212005038848001778112001619904384
7628812829548147233283064161635259750442066781200593534407684356374400013005619200177972258688

All 15 rows

Technical details
Node id
n-a148d5351dbfaee8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9b3fddac6ffcbf8c

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:           8,           80,            108;
  3:          56,          944,           2916,           2304;
  4:         616,        14864,          74556,         129024,           72000;
  5:        9856,       311744,        2196216,        6040704,         7200000,         3110400;
  6:      216832,      8514176,       77776560,      296879616,       554821200,       503884800,        177811200;
  7:     6288128,    295481472,     3328306416,    16352597504,     42066781200,     59353440768,      43563744000,      13005619200;
  8:   232660736,  12743795328,   171015335856,  1031069328896,   3396138123600,   6557561183232,    7428495723264,    4577977958400,   1185137049600;
  9: 10702393856, 669972450048, 10447324003296, 74791642866176, 301216478061600, 741787315247232, 1145012048216064, 1083814238453760, 575976606105600, 131681894400000;
%F Formula
The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(8*n^5-8*n^4+8*n^3+8*n^2-16*n)*x-(36*n^5-36*n^4-36*n^3+36*n^2)*x^2)*P(n-2,x) + ((8*n^5-8*n^4+8*n^3+8*n^2-16*n)*x^2+(16*n^5-16*n^4-16*n^3+16*n^2)*x^3)*d/dx P(n-2,x) + (4*n^3+4*n+(16*n^3-16*n)*x)*P(n-1,x) - (16*n^3-16*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x - (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^2)*P[n - 2, x] + ((8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x^2 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*x^3)*D[P[n - 2, x], x] + (4*n^3 + 4*n + (16*n^3 - 16*n)*x)*P[n - 1, x] - (16*n^3 - 16*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x - (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^2)*v[i - 2] + ((8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x^2 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*x^3)*deriv(v[i - 2], x) + (4*n^3 + 4*n + (16*n^3 - 16*n)*x)*v[i - 1] - (16*n^3 - 16*n)*x^2*deriv(v[i - 1], x))/(4*n - 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. [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.