Real-rooted atlas

A153653: Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, read by rows.

A153653 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A153653

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-646\,x + 323\,n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 19 + 19\,x\)
\(P_{2}(x) = 2 + 718\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1191938
227182722
3268576857213718
42750524562875052260642
52815324679442467944815324952198
62880149002128427373249002128801294091762
72944975424328863198568863198567542432944921787743478

All 15 rows b-files: A153653, A153653 (reversed triangle) ?

Technical details
Node id
n-71e25d00553f2c01 · revision 0 · JSON
Definition source
oeis_catalog:A153653
Fingerprint
f6-fac9c3c65d64b266

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: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

  • A153653 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
  • A153653 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  2;
  1: 19,    19;
  2:  2,   718,        2;
  3:  2,  6857,     6857,          2;
  4:  2,  7505,   245628,       7505,            2;
  5:  2,  8153,  2467944,    2467944,         8153,            2;
  6:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;
  7:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;
  8:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;
  9:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (323*n-646)*x*P(n-2,x) for n >= 3, with P(0,x) = 2, P(1,x) = 19 + 19*x, P(2,x) = 2 + 718*x + 2*x^2.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 19 + 19*x;
P[2, x_] := 2 + 718*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (323*n - 646)*x*P[n - 2, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (323*n - 646)*x*v[i - 2])/(n - 2)); 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

A153653 triangle, offset 1, from P0
%e Example
Triangle begins:
   1:  2;
   2: 19,    19;
   3:  2,   718,        2;
   4:  2,  6857,     6857,          2;
   5:  2,  7505,   245628,       7505,            2;
   6:  2,  8153,  2467944,    2467944,         8153,            2;
   7:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;
   8:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;
   9:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;
  10:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (323*n-969)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 19 + 19*x, P(3,x) = 2 + 718*x + 2*x^2.
%t Mathematica
P[1, x_] := 2;
P[2, x_] := 19 + 19*x;
P[3, x_] := 2 + 718*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (323*n - 969)*x*P[n - 2, x])/(n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (323*n - 969)*x*v[i - 2])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A153653 reversed triangle, offset 1, from P0
%e Example
Triangle begins:
   1:  2;
   2: 19,    19;
   3:  2,   718,        2;
   4:  2,  6857,     6857,          2;
   5:  2,  7505,   245628,       7505,            2;
   6:  2,  8153,  2467944,    2467944,         8153,            2;
   7:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;
   8:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;
   9:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;
  10:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (323*n-969)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 19 + 19*x, P(3,x) = 2 + 718*x + 2*x^2.
%t Mathematica
P[1, x_] := 2;
P[2, x_] := 19 + 19*x;
P[3, x_] := 2 + 718*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (323*n - 969)*x*P[n - 2, x])/(n - 3)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (323*n - 969)*x*v[i - 2])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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