Real-rooted atlas

A316649: Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.

A316649 ↗ nonnegative coefficients

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

Recurrence from A316649

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} - 5\,n - 15\,n x - 10\,n x^{2} + 2\,n^{2} + 6\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2 - 4\,x - 2\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = x + 2\,x^{2}\)
\(P_{2}(x) = x + 7\,x^{2} + 12\,x^{3} + 6\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
10123
201712626
3011455927020252
40123153471780720350702568
5013433615844251700272384592163825226928
6014764042101617539733652267266054390262507392924287648
7016211079596492251648983807316235767326186147003601621401843260434323112896

All 15 rows b-files: A052141 (row sums), A316649 ?

Technical details
Node id
n-97a17a1dc7998ff8 · revision 0 · JSON
Definition source
oeis_catalog:A316649
Fingerprint
f6-bdbbd88be9205817

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

  • A316649 triangle 64 terms, 8 rows · tabf · nonn · catalog b-file?

Row sums ? A052141

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 1,  2;
  2: 0, 1,  7,   12,     6;
  3: 0, 1, 14,   55,    92,     70,      20;
  4: 0, 1, 23,  153,   471,    780,     720,     350,       70;
  5: 0, 1, 34,  336,  1584,   4251,    7002,    7238,     4592,     1638,      252;
  6: 0, 1, 47,  640,  4210,  16175,   39733,   65226,    72660,    54390,    26250,     7392,      924;
  7: 0, 1, 62, 1107,  9596,  49225,  164898,  380731,   623576,   732618,   614700,   360162,   140184,    32604,     3432;
  8: 0, 1, 79, 1785, 19607, 128345,  555303, 1685257,  3716695,  6077196,  7430220,  6780312,  4557168,  2192190,   714714,   141570,   12870;
  9: 0, 1, 98, 2728, 36896, 298480, 1607648, 6139504, 17290304, 36808723, 60110030, 75805972, 73788176, 54980926, 30802772, 12563980, 3523520, 607750, 48620;
%F Formula
The row polynomials satisfy (n^2-2*n)*P(n,x) = (2*n^2-5*n+2+(6*n^2-15*n+6)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) - (n^2-3*n+2+(2*n^2-6*n+4)*x+(n^2-3*n+2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x + 2*x^2, P(2,x) = x + 7*x^2 + 12*x^3 + 6*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[2, x_] := x + 7*x^2 + 12*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x])/(n^2 - 2*n)]];
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] = x + 2*x^2; v[3] = x + 7*x^2 + 12*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2])/(n^2 - 2*n)); 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

A316649 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 0, 1,  2;
  2: 0, 1,  7,   12,     6;
  3: 0, 1, 14,   55,    92,     70,      20;
  4: 0, 1, 23,  153,   471,    780,     720,     350,       70;
  5: 0, 1, 34,  336,  1584,   4251,    7002,    7238,     4592,     1638,      252;
  6: 0, 1, 47,  640,  4210,  16175,   39733,   65226,    72660,    54390,    26250,     7392,      924;
  7: 0, 1, 62, 1107,  9596,  49225,  164898,  380731,   623576,   732618,   614700,   360162,   140184,    32604,     3432;
  8: 0, 1, 79, 1785, 19607, 128345,  555303, 1685257,  3716695,  6077196,  7430220,  6780312,  4557168,  2192190,   714714,   141570,   12870;
  9: 0, 1, 98, 2728, 36896, 298480, 1607648, 6139504, 17290304, 36808723, 60110030, 75805972, 73788176, 54980926, 30802772, 12563980, 3523520, 607750, 48620;
%F Formula
The row polynomials satisfy (n^2-2*n)*P(n,x) = (2*n^2-5*n+2+(6*n^2-15*n+6)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) - (n^2-3*n+2+(2*n^2-6*n+4)*x+(n^2-3*n+2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x + 2*x^2, P(2,x) = x + 7*x^2 + 12*x^3 + 6*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[2, x_] := x + 7*x^2 + 12*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x])/(n^2 - 2*n)]];
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] = x + 2*x^2; v[3] = x + 7*x^2 + 12*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1] - (n^2 - 3*n + 2 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2])/(n^2 - 2*n)); 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

  • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 5

References

  1. [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