A052553 → basis A013609 → basis A013609
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(3 + 4\,x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 1 | |||||||
| 3 | 4 | 4 | 8 | ||||||
| 4 | 7 | 8 | 15 | ||||||
| 5 | 19 | 36 | 16 | 71 | |||||
| 6 | 40 | 88 | 48 | 176 | |||||
| 7 | 97 | 272 | 240 | 64 | 673 | ||||
| 8 | 217 | 696 | 736 | 256 | 1905 | ||||
| 9 | 508 | 1900 | 2544 | 1408 | 256 | 6616 | |||
| 10 | 1159 | 4856 | 7536 | 5120 | 1280 | 19951 | |||
| 11 | 2683 | 12588 | 22768 | 19520 | 7680 | 1024 | 66263 | ||
| 12 | 6160 | 31792 | 64800 | 65024 | 32000 | 6144 | 205920 | ||
| 13 | 14209 | 80288 | 183456 | 214656 | 133120 | 39936 | 4096 | 669761 | |
| 14 | 32689 | 200304 | 505024 | 668928 | 489216 | 186368 | 28672 | 2111201 | |
| 15 | 75316 | 498004 | 1376544 | 2046720 | 1747200 | 838656 | 200704 | 16384 | 6799528 |
Compute more
?
Technical details
- Node id
n-44317ac492bee6a1· revision 0 · JSON- Definition source
derived:basis_map_A013609- Fingerprint
f6-ffa46b3d424e7d6a
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: 4, 4; 4: 7, 8; 5: 19, 36, 16; 6: 40, 88, 48; 7: 97, 272, 240, 64; 8: 217, 696, 736, 256; 9: 508, 1900, 2544, 1408, 256; 10: 1159, 4856, 7536, 5120, 1280;
%F FormulaThe row polynomials satisfy P(n,x) = (3+4*x)*P(n-2,x) + P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + 4*x)*P[n - 2, x] + P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (3 + 4*x)*v[i - 2] + v[i - 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 (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antid…?
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Geometric basis map (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)
-
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 ›
-
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 ›
-
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 ›
-
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.?
-
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 ›
-
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 ›
-
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
-
-
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.?
-
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.?
-
Geometric basis map (elementary) polytool ? step ›
-
-
from
basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antid…?
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
- [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