A308737: Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.
A308737 ↗ nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A308737
\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-2 - 4\,x - 2\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 3 | 4 | ||||||
| 2 | 1 | 8 | 7 | 16 | |||||
| 3 | 1 | 17 | 31 | 15 | 64 | ||||
| 4 | 1 | 34 | 96 | 94 | 31 | 256 | |||
| 5 | 1 | 67 | 258 | 382 | 253 | 63 | 1024 | ||
| 6 | 1 | 132 | 645 | 1280 | 1275 | 636 | 127 | 4096 | |
| 7 | 1 | 261 | 1545 | 3845 | 5115 | 3831 | 1531 | 255 | 16384 |
All 15 rows b-files: A000302 (row sums), A308737 ?
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Technical details
- Node id
n-13e23eca5f474574· revision 0 · JSON- Definition source
oeis_catalog:A308737- Fingerprint
f6-6fbba7c3cf69e7d1
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): yes
Matches ?
Row sums ? A155543, A347478, A000302
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 3; 2: 1, 8, 7; 3: 1, 17, 31, 15; 4: 1, 34, 96, 94, 31; 5: 1, 67, 258, 382, 253, 63; 6: 1, 132, 645, 1280, 1275, 636, 127; 7: 1, 261, 1545, 3845, 5115, 3831, 1531, 255; 8: 1, 518, 3598, 10766, 17920, 17906, 10738, 3578, 511; 9: 1, 1031, 8212, 28700, 57358, 71666, 57316, 28652, 8185, 1023;
%F FormulaThe row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, 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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 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
A308737 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 3; 2: 1, 8, 7; 3: 1, 17, 31, 15; 4: 1, 34, 96, 94, 31; 5: 1, 67, 258, 382, 253, 63; 6: 1, 132, 645, 1280, 1275, 636, 127; 7: 1, 261, 1545, 3845, 5115, 3831, 1531, 255; 8: 1, 518, 3598, 10766, 17920, 17906, 10738, 3578, 511; 9: 1, 1031, 8212, 28700, 57358, 71666, 57316, 28652, 8185, 1023;
%F FormulaThe row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 (−∞, 0)
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
Proof chains ? JSON
Used in 11
- basis_map_A013611 of Triangle of scaled 1-tiered binomial coefficients, T(n,k) … Image under a basis map u^k -> P_k (A013611)
- basis_map_A013610 of Triangle of scaled 1-tiered binomial coefficients, T(n,k) … Image under a basis map u^k -> P_k (A013610)
- basis_map_A013609 of Triangle of scaled 1-tiered binomial coefficients, T(n,k) … Image under a basis map u^k -> P_k (A013609)
- basis_map_A008277 of Triangle of scaled 1-tiered binomial coefficients, T(n,k) … Image under a basis map u^k -> P_k (A008277)
- shift_x_plus_one of Triangle of scaled 1-tiered binomial coefficients, T(n,k) =… Image under a basis map u^k -> P_k (A007318)
- mobius_x_over_x_plus_one of Triangle of scaled 1-tiered binomial coefficients, … Image under a real-rootedness preserver (mobius, shift 0)
- one_plus_derivative of Triangle of scaled 1-tiered binomial coefficients, T(n,k… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- shift_x_plus_one of Triangle of scaled 1-tiered binomial coefficients, T(n,k) =… Image under a real-rootedness preserver (affine, shift 0)
3 more
- reversal of Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)… Image under a real-rootedness preserver (reversal, shift 0)
- theta_plus_one of Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2… Image under a real-rootedness preserver (euler polynomial, shift 0)
- derivative of Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+… Image under a real-rootedness preserver (derivative polynomial, shift 0)
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