A052553 → x ↦ x+1 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-9 - 4\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-18 - 17\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 1 | |||||||
| 3 | 3 | 1 | 4 | ||||||
| 4 | 5 | 2 | 7 | ||||||
| 5 | 11 | 7 | 1 | 19 | |||||
| 6 | 21 | 16 | 3 | 40 | |||||
| 7 | 43 | 41 | 12 | 1 | 97 | ||||
| 8 | 85 | 94 | 34 | 4 | 217 | ||||
| 9 | 171 | 219 | 99 | 18 | 1 | 508 | |||
| 10 | 341 | 492 | 261 | 60 | 5 | 1159 | |||
| 11 | 683 | 1101 | 678 | 195 | 25 | 1 | 2683 | ||
| 12 | 1365 | 2426 | 1692 | 576 | 95 | 6 | 6160 | ||
| 13 | 2731 | 5311 | 4149 | 1644 | 340 | 33 | 1 | 14209 | |
| 14 | 5461 | 11528 | 9959 | 4488 | 1106 | 140 | 7 | 32689 | |
| 15 | 10923 | 24881 | 23568 | 11925 | 3430 | 546 | 42 | 1 | 75316 |
Compute more
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Technical details
- Node id
n-8f1dfaae790c20d9· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-3d8013fbbe82cb22
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1; 3: 3, 1; 4: 5, 2; 5: 11, 7, 1; 6: 21, 16, 3; 7: 43, 41, 12, 1; 8: 85, 94, 34, 4; 9: 171, 219, 99, 18, 1; 10: 341, 492, 261, 60, 5;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (5*n-9+(2*n-4)*x)*P(n-1,x) - (18+17*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 9 + (2*n - 4)*x)*P[n - 1, x] - (18 + 17*x + 4*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((5*n - 9 + (2*n - 4)*x)*v[i - 1] - (18 + 17*x + 4*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); 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 ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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
shift_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidi…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
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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 ›
-
from
Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
- ↑ 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
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
no recorded property covers this requirement
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
no recorded property covers this requirement
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
-
from
Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
from
Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
-
from
Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
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
-
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 ›
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
-
from
delannoy?
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
from
delannoy?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
-
from
delannoy?
-
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…?
-
Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
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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) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
-
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
no recorded property covers this requirement
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
- ↑ Ma–Wang interval theorem (above)
-
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
-
from
shift_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidi…?
Related ?
Same fingerprint ?
- A249139: Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
- A111006 → x ↦ x/(x+1)
- A128100 → basis A007318
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