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 ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 2 | 2 | |||||||
| 1 | 3 | 3 | 6 | ||||||
| 2 | 2 | 14 | 2 | 18 | |||||
| 3 | 2 | 25 | 25 | 2 | 54 | ||||
| 4 | 2 | 33 | 92 | 33 | 2 | 162 | |||
| 5 | 2 | 41 | 200 | 200 | 41 | 2 | 486 | ||
| 6 | 2 | 49 | 340 | 676 | 340 | 49 | 2 | 1458 | |
| 7 | 2 | 57 | 512 | 1616 | 1616 | 512 | 57 | 2 | 4374 |
All 15 rows b-files: A008776 (row sums), A153516, A153516 (reversed triangle) ?
Compute more
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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 ExampleTriangle 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 FormulaThe 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 MathematicaP[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/GPP(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 ExampleTriangle 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 FormulaThe 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 MathematicaP[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/GPP(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 ExampleTriangle 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 FormulaRow 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 MathematicaP[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/GPP(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
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 2
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
Proof chains ? JSON
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
References
- [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