basis-map coefficients of A026729 → basis A013610 → basis A007318
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 4 | 3 | 7 | ||||||
| 3 | 16 | 24 | 9 | 49 | |||||
| 4 | 64 | 144 | 108 | 27 | 343 | ||||
| 5 | 256 | 768 | 864 | 432 | 81 | 2401 | |||
| 6 | 1024 | 3840 | 5760 | 4320 | 1620 | 243 | 16807 | ||
| 7 | 4096 | 18432 | 34560 | 34560 | 19440 | 5832 | 729 | 117649 | |
| 8 | 16384 | 86016 | 193536 | 241920 | 181440 | 81648 | 20412 | 2187 | 823543 |
Compute more
?
Technical details
- Node id
n-059009e413050919· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-c91d5d506c107735
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 -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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 4, 3; 3: 16, 24, 9; 4: 64, 144, 108, 27; 5: 256, 768, 864, 432, 81; 6: 1024, 3840, 5760, 4320, 1620, 243; 7: 4096, 18432, 34560, 34560, 19440, 5832, 729; 8: 16384, 86016, 193536, 241920, 181440, 81648, 20412, 2187; 9: 65536, 393216, 1032192, 1548288, 1451520, 870912, 326592, 69984, 6561; 10: 262144, 1769472, 5308416, 9289728, 10450944, 7838208, 3919104, 1259712, 236196, 19683;
%F FormulaThe row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + 3*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; for(n = 2, N, my(i = n); v[i] = (4 + 3*x)*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 (−∞, -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)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
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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
basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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from
veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
-
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
-
from
veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
-
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
-
from
veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
-
from
shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
-
from
basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
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from
basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
Related ?
Same fingerprint ?
- A356546 → x ↦ x+1 → basis A013611
- A038329 → basis A013610 → basis A013609
- basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
- A038284 → reverse
- A156992 → basis A013611 → basis A013609
- A038330 → basis A013610 → D
- A178822 → x ↦ x+1 → basis A013611
- A038335 → D
- A178820 → basis A007318 → basis A013611
- A003506 → x ↦ x+1 → basis A013611
- A120909 → basis A013610 → basis A013609
- A038284 → D
- A085880 → x ↦ x+1 → basis A013611
- A085881 → basis A013611 → basis A013609
- A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
- A038221 → basis A013611 → basis A013609
- A130123 → basis A013610 → x ↦ x+1
- A178821 → basis A007318 → basis A013611
- basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
- A094305 → x ↦ x+1 → basis A013611
- basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
- basis-map coefficients of A026729 → basis A013609 → basis A013609
- A120910 → basis A013611 → D
- A196347 → basis A013611 → basis A013609
- A038208 → basis A013611 → basis A013609
- basis-map coefficients of A112227 → basis A013609 → basis A013609
- A120910 → D → basis A013611
- A013610 → basis A007318
- A178821 → basis A013611 → basis A013609
- A038329 → basis A013611 → basis A013610
- A013609 → D → basis A013609
- A038330 → basis A013610
- A013610 → basis A013611 → x ↦ x+1
- A120910 → basis A013611
- A038335 → reverse
- A130123 → basis A013609 → basis A013609
- A136572 → basis A013610 → basis A007318
- A038334 → basis A013609 → basis A013610
- A013610 → basis A007318 → D
- A038279 → basis A013610 → x ↦ x/(x+1)
- A038243 → basis A013609 → x ↦ x/(x+1)
- A356546 → basis A013611 → basis A013609
- A178820 → basis A013611 → basis A013609
- A027465 → D → basis A013610
- A013609 → basis A013609 → D
- A178822 → basis A013611 → basis A013609
- A196347 → x ↦ x+1 → basis A013611
- A013609 → basis A013609
- Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
- A135065 → x ↦ x+1 → basis A013611
- A120910 → basis A013611 → reverse
- A085880 → basis A013611 → basis A013609
- A027465 → x ↦ x/(x+1)
- basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
- basis-map coefficients of A112227 → basis A013610 → basis A007318
- A028326 → basis A013611 → basis A013609
- A038330 → basis A013610 → reverse
- A028326 → basis A007318 → basis A013611
- A038330 → x ↦ x/(x+1)
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.