A144443 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(6 + 6\,x\right)\,P_{n-2}(x) + \left(-4 - 5\,x + 6\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 22 | 22 | 1 | 45 | |||||
| 3 | 320 | 480 | 162 | 1 | 963 | ||||
| 4 | 6532 | 13064 | 7674 | 1142 | 1 | 28413 | |||
| 5 | 171752 | 429380 | 359504 | 109876 | 8002 | 1 | 1078515 | ||
| 6 | 5535256 | 16605768 | 18049008 | 8421736 | 1499262 | 56022 | 1 | 50167053 | |
| 7 | 211370240 | 739795840 | 988786320 | 622476200 | 181709920 | 19986600 | 392162 | 1 | 2764517283 |
Compute more
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Technical details
- Node id
n-9109503bbc9c6251· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-4686d3e0771fa802
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 ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 22, 22, 1; 3: 320, 480, 162, 1; 4: 6532, 13064, 7674, 1142, 1; 5: 171752, 429380, 359504, 109876, 8002, 1; 6: 5535256, 16605768, 18049008, 8421736, 1499262, 56022, 1; 7: 211370240, 739795840, 988786320, 622476200, 181709920, 19986600, 392162, 1; 8: 9333502096, 37334008384, 59221536976, 46995581584, 19255629388, 3741632584, 263299234, 2745142, 1; 9: 467943326240, 2105744968080, 3866152258560, 3704723053920, 1962334509912, 557433565500, 74899480944, 3447204156, 19216002, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (6+6*x)*P(n-2,x) + (6*n-4+(6*n-5)*x)*P(n-1,x) - (6*x+6*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(6 + 6*x)*P[n - 2, x] + (6*n - 4 + (6*n - 5)*x)*P[n - 1, x] - (6*x + 6*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (6 + 6*x)*v[i - 2] + (6*n - 4 + (6*n - 5)*x)*v[i - 1] - (6*x + 6*x^2)*deriv(v[i - 1], x)); 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
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
- 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 (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
Related ?
Same fingerprint ?
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