A249139 → basis A013611 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(4 + 4\,n\right)\,P_{n}(x) = \left(4 + 60\,n + 32\,n x\right)\,P_{n-1}(x) + \left(-203 - 228\,x - 64\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 8 | 4 | 12 | ||||||
| 2 | 15 | 8 | 23 | ||||||
| 3 | 71 | 68 | 16 | 155 | |||||
| 4 | 176 | 184 | 48 | 408 | |||||
| 5 | 673 | 944 | 432 | 64 | 2113 | ||||
| 6 | 1905 | 2936 | 1504 | 256 | 6601 | ||||
| 7 | 6616 | 12236 | 8304 | 2432 | 256 | 29844 | |||
| 8 | 19951 | 40408 | 30576 | 10240 | 1280 | 102455 | |||
| 9 | 66263 | 152524 | 137648 | 60480 | 12800 | 1024 | 430739 | ||
| 10 | 205920 | 515184 | 513312 | 254464 | 62720 | 6144 | 1557744 | ||
| 11 | 669761 | 1847904 | 2086944 | 1228416 | 394240 | 64512 | 4096 | 6295873 | |
| 12 | 2111201 | 6277872 | 7740864 | 5062912 | 1851136 | 358400 | 28672 | 23431057 | |
| 13 | 6799528 | 21892244 | 29741088 | 22009600 | 9524480 | 2386944 | 315392 | 16384 | 92685660 |
| 14 | 21577935 | 74282152 | 109038624 | 88413440 | 42734080 | 12300288 | 1949696 | 131072 | 350427287 |
Compute more
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Technical details
- Node id
n-244806c82723e6d2· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-9395ebb6ef054cce
Observed on the stored rows ?
- Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
- 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: 8, 4; 2: 15, 8; 3: 71, 68, 16; 4: 176, 184, 48; 5: 673, 944, 432, 64; 6: 1905, 2936, 1504, 256; 7: 6616, 12236, 8304, 2432, 256; 8: 19951, 40408, 30576, 10240, 1280; 9: 66263, 152524, 137648, 60480, 12800, 1024;
%F FormulaThe row polynomials satisfy (4*n+4)*P(n,x) = (60*n+4+32*n*x)*P(n-1,x) - (203+228*x+64*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 4 + 32*n*x)*P[n - 1, x] - (203 + 228*x + 64*x^2)*D[P[n - 1, x], x])/(4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 4 + 32*n*x)*v[i - 1] - (203 + 228*x + 64*x^2)*deriv(v[i - 1], x))/(4*n + 4)); 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)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A013611 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Co…?
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
delannoy?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
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from
delannoy?
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
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from
delannoy?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
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from
delannoy?
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from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
- ↑ Ma–Wang interval theorem (above)
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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from
Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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from
shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
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from
Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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from
basis_map_A013611 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Co…?
Related ?
Candidate equalities ?
- A249139 → basis A013611 → basis A007318 15 rows; no all-index equality certificate was found
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
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741