A146986: Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
A146986 ↗ palindromicγ-positivenonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A146986
\(\left(-3 + n\right)\,P_{n}(x) = \left(-9 - 9\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(6 + 12\,x + 6\,x^{2} - 2\,n - 4\,n x - 2\,n x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
\(P_{2}(x) = 1 + 4\,x + x^{2}\)
\(P_{3}(x) = 1 + 7\,x + 7\,x^{2} + x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 1 | 2 | |||||||||||||
| 2 | 1 | 4 | 1 | 6 | ||||||||||||
| 3 | 1 | 7 | 7 | 1 | 16 | |||||||||||
| 4 | 1 | 12 | 22 | 12 | 1 | 48 | ||||||||||
| 5 | 1 | 21 | 58 | 58 | 21 | 1 | 160 | |||||||||
| 6 | 1 | 38 | 143 | 212 | 143 | 38 | 1 | 576 | ||||||||
| 7 | 1 | 71 | 341 | 675 | 675 | 341 | 71 | 1 | 2176 | |||||||
| 8 | 1 | 136 | 796 | 1976 | 2630 | 1976 | 796 | 136 | 1 | 8448 | ||||||
| 9 | 1 | 265 | 1828 | 5460 | 9086 | 9086 | 5460 | 1828 | 265 | 1 | 33280 | |||||
| 10 | 1 | 522 | 4141 | 14456 | 28882 | 36092 | 28882 | 14456 | 4141 | 522 | 1 | 132096 | ||||
| 11 | 1 | 1035 | 9271 | 37029 | 86346 | 129486 | 129486 | 86346 | 37029 | 9271 | 1035 | 1 | 526336 | |||
| 12 | 1 | 2060 | 20546 | 92380 | 246255 | 430872 | 517020 | 430872 | 246255 | 92380 | 20546 | 2060 | 1 | 2101248 | ||
| 13 | 1 | 4109 | 45134 | 225566 | 676555 | 1352967 | 1894068 | 1894068 | 1352967 | 676555 | 225566 | 45134 | 4109 | 1 | 8396800 | |
| 14 | 1 | 8206 | 98395 | 541036 | 1803241 | 4057042 | 6491067 | 7572840 | 6491067 | 4057042 | 1803241 | 541036 | 98395 | 8206 | 1 | 33570816 |
b-files: A146986, A146986 (reversed triangle) ?
Compute more
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Technical details
- Node id
n-02236a48f5253afe· revision 0 · JSON- Definition source
oeis_catalog:A146986- Fingerprint
f6-a5de81f788c8c901
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 ?
- A146986 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
- A146986 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 4, 1; 3: 1, 7, 7, 1; 4: 1, 12, 22, 12, 1; 5: 1, 21, 58, 58, 21, 1; 6: 1, 38, 143, 212, 143, 38, 1; 7: 1, 71, 341, 675, 675, 341, 71, 1; 8: 1, 136, 796, 1976, 2630, 1976, 796, 136, 1; 9: 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1;
%F FormulaThe row polynomials satisfy (n-3)*P(n,x) = (3*n-9+(3*n-9)*x)*P(n-1,x) - (2*n-6+(4*n-12)*x+(2*n-6)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2, P(3,x) = 1 + 7*x + 7*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[3, x_] := 1 + 7*x + 7*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 9 + (3*n - 9)*x)*P[n - 1, x] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*P[n - 2, x])/(n - 3)]];
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 + x; v[3] = 1 + 4*x + x^2; v[4] = 1 + 7*x + 7*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((3*n - 9 + (3*n - 9)*x)*v[i - 1] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*v[i - 2])/(n - 3)); 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
A146986 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 4, 1; 3: 1, 7, 7, 1; 4: 1, 12, 22, 12, 1; 5: 1, 21, 58, 58, 21, 1; 6: 1, 38, 143, 212, 143, 38, 1; 7: 1, 71, 341, 675, 675, 341, 71, 1; 8: 1, 136, 796, 1976, 2630, 1976, 796, 136, 1; 9: 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1;
%F FormulaThe row polynomials satisfy (n-3)*P(n,x) = (3*n-9+(3*n-9)*x)*P(n-1,x) - (2*n-6+(4*n-12)*x+(2*n-6)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2, P(3,x) = 1 + 7*x + 7*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[3, x_] := 1 + 7*x + 7*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 9 + (3*n - 9)*x)*P[n - 1, x] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*P[n - 2, x])/(n - 3)]];
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 + x; v[3] = 1 + 4*x + x^2; v[4] = 1 + 7*x + 7*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((3*n - 9 + (3*n - 9)*x)*v[i - 1] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*v[i - 2])/(n - 3)); 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
A146986 reversed triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 4, 1; 3: 1, 7, 7, 1; 4: 1, 12, 22, 12, 1; 5: 1, 21, 58, 58, 21, 1; 6: 1, 38, 143, 212, 143, 38, 1; 7: 1, 71, 341, 675, 675, 341, 71, 1; 8: 1, 136, 796, 1976, 2630, 1976, 796, 136, 1; 9: 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (3*n-9+(3*n-9)*x)*P(n-1,x) - (2*n-6+(4*n-12)*x+(2*n-6)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2, P(3,x) = 1 + 7*x + 7*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[3, x_] := 1 + 7*x + 7*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 9 + (3*n - 9)*x)*P[n - 1, x] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*P[n - 2, x])/(n - 3)]];
Table[Reverse[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 + x; v[3] = 1 + 4*x + x^2; v[4] = 1 + 7*x + 7*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((3*n - 9 + (3*n - 9)*x)*v[i - 1] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*v[i - 2])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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 ≥ 3
Proof chains ? JSON
Used in 5
- basis_map_A013611 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2… Image under a basis map u^k -> P_k (A013611)
- basis_map_A013609 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2… Image under a basis map u^k -> P_k (A013609)
- basis_map_A008277 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2… Image under a basis map u^k -> P_k (A008277)
- basis_map_A007318 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2… Image under a basis map u^k -> P_k (A007318)
- basis_map_A013610 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2… Image under a basis map u^k -> P_k (A013610)
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