Real-rooted atlas

A153516: Triangle T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j) and (p,q,j) = (0,1,2), read by rows.

A153516 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A153516

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-6\,x + 3\,n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 3 + 3\,x\)
\(P_{2}(x) = 2 + 14\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1336
2214218
322525254
423392332162
5241200200412486
62493406763404921458
7257512161616165125724374
826571631485260314871665213122
927395254001325613256540095273239366
102811220850028100422922810085001220812118098
112891520125765280011016011016052800125761520892354294
1229718521775690876247260347196247260908761775618529721062882
13210522162416814636049653692493692493649653614636024168221610523188646
14211326123194022379691552421632522891460216325291552422379631940261211329565938

b-files: A008776 (row sums), A153516, A153516 (reversed triangle) ?

Technical details
Node id
n-71548025909043e7 · revision 0 · JSON
Definition source
oeis_catalog:A153516
Fingerprint
f6-54adaf1b1c7368ef

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 ?

  • A153516 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
  • A153516 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A110593, A027334, A025192, A099856, A008776

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 2;
  1: 3,  3;
  2: 2, 14,   2;
  3: 2, 25,  25,    2;
  4: 2, 33,  92,   33,     2;
  5: 2, 41, 200,  200,    41,     2;
  6: 2, 49, 340,  676,   340,    49,    2;
  7: 2, 57, 512, 1616,  1616,   512,   57,   2;
  8: 2, 65, 716, 3148,  5260,  3148,  716,  65,  2;
  9: 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (3*n-6)*x*P(n-2,x) for n >= 3, with P(0,x) = 2, P(1,x) = 3 + 3*x, P(2,x) = 2 + 14*x + 2*x^2.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 3*x;
P[2, x_] := 2 + 14*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (3*n - 6)*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] = 3 + 3*x; v[3] = 2 + 14*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (3*n - 6)*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

A153516 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 2;
   2: 3,  3;
   3: 2, 14,   2;
   4: 2, 25,  25,    2;
   5: 2, 33,  92,   33,     2;
   6: 2, 41, 200,  200,    41,     2;
   7: 2, 49, 340,  676,   340,    49,    2;
   8: 2, 57, 512, 1616,  1616,   512,   57,   2;
   9: 2, 65, 716, 3148,  5260,  3148,  716,  65,  2;
  10: 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (3*n-9)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 3 + 3*x, P(3,x) = 2 + 14*x + 2*x^2.
%t Mathematica
P[1, x_] := 2;
P[2, x_] := 3 + 3*x;
P[3, x_] := 2 + 14*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (3*n - 9)*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] = 3 + 3*x; v[3] = 2 + 14*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (3*n - 9)*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

A153516 reversed triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 2;
   2: 3,  3;
   3: 2, 14,   2;
   4: 2, 25,  25,    2;
   5: 2, 33,  92,   33,     2;
   6: 2, 41, 200,  200,    41,     2;
   7: 2, 49, 340,  676,   340,    49,    2;
   8: 2, 57, 512, 1616,  1616,   512,   57,   2;
   9: 2, 65, 716, 3148,  5260,  3148,  716,  65,  2;
  10: 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 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) + (3*n-9)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 3 + 3*x, P(3,x) = 2 + 14*x + 2*x^2.
%t Mathematica
P[1, x_] := 2;
P[2, x_] := 3 + 3*x;
P[3, x_] := 2 + 14*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (3*n - 9)*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] = 3 + 3*x; v[3] = 2 + 14*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (3*n - 9)*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