basis-map coefficients of A026729 → basis A013611 → D
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-2 + n\right)\,P_{n}(x) = \left(-1 - 4\,x + n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0 | |||||||
| 2 | 4 | 4 | ||||||
| 3 | 8 | 32 | 40 | |||||
| 4 | 12 | 96 | 192 | 300 | ||||
| 5 | 16 | 192 | 768 | 1024 | 2000 | |||
| 6 | 20 | 320 | 1920 | 5120 | 5120 | 12500 | ||
| 7 | 24 | 480 | 3840 | 15360 | 30720 | 24576 | 75000 | |
| 8 | 28 | 672 | 6720 | 35840 | 107520 | 172032 | 114688 | 437500 |
Compute more
?
Technical details
- Node id
n-be86902a3ebfde1b· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-03d7b8417365c509
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
- 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: 0; 2: 4; 3: 8, 32; 4: 12, 96, 192; 5: 16, 192, 768, 1024; 6: 20, 320, 1920, 5120, 5120; 7: 24, 480, 3840, 15360, 30720, 24576; 8: 28, 672, 6720, 35840, 107520, 172032, 114688; 9: 32, 896, 10752, 71680, 286720, 688128, 917504, 524288; 10: 36, 1152, 16128, 129024, 645120, 2064384, 4128768, 4718592, 2359296;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (n-1+(4*n-4)*x)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t MathematicaP[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (4*n - 4)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((n - 1 + (4*n - 4)*x)*v[i - 1])/(n - 2)); 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 Laguerre’s theorem on differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
basis_map_A013611 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_A013610 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
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).?
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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
-
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).?
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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
-
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 >…?
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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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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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from
basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
Related ?
Same fingerprint ?
- A131436 → basis A013611
- image under θ+1 → (θ+1)² → basis A013611
- basis-map coefficients of A021012 → basis A013611
- A013610 → basis A013611 → x ↦ x/(x+1)
- A038303 → basis A007318 → basis A013610
- basis-map coefficients of A026729 → D → basis A013611
- A130123 → D → basis A013611
- A131431 → basis A013611
- A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
- basis-map coefficients of A026729 → basis A013611 → reverse
- basis-map coefficients of A112227 → θ+1 → basis A013611
- A013611 → D
- A038330 → basis A007318 → reverse
- basis-map coefficients of A026729 → (θ+1)² → basis A013611
- A027466 → basis A013609 → D
- basis-map coefficients of A026729 → basis A013611
- A130123 → basis A013611 → D
- basis-map coefficients of A030528, shifted → D → basis A013611
- A136572 → (θ+1)² → basis A013611
- basis-map coefficients of A030528, shifted → basis A013611
- A136572 → basis A013611 → reverse
- A038330 → basis A007318 → D
- A136572 → basis A013611
- A130123 → basis A013611
- A136572 → D → basis A013611
- A038231 → D
- A136572 → basis A013611 → D
- A130123 → θ+1 → basis A013611
- basis-map coefficients of A026729, shifted → basis A013611
- basis-map coefficients of A026729, shifted → D → basis A013611
- A134309 → basis A013611
- A132440 → θ+1 → basis A013611
- A136572 → θ+1 → basis A013611
- A027467 → basis A013611 → D
- A131431 → D → basis A013611
- basis-map coefficients of A030528, shifted → θ+1 → basis A013611
- image under θ+1 → θ+1 → D → basis A013611
- basis-map coefficients of A112227 → basis A013611
- A038231 → D → D
- A127648 → D → basis A013611
- A136521 → basis A013611
- A132440 → basis A013611
- basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
- A013611: Triangle of coefficients in expansion of (1+4x)^n.
- basis-map coefficients of A112227 → basis A013611 → D
- A129186 → basis A013611
- A131431 → θ+1 → basis A013611
- image under θ+1 → basis A013611
- basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
- A120910 → basis A007318 → basis A007318
- A027466 → basis A013609
- A038330 → basis A007318
- A038329 → reverse
- A038329 → D
- A130123 → (θ+1)² → basis A013611
- A027467 → basis A013611 → reverse
- basis-map coefficients of A026729, shifted → θ+1 → basis A013611
- basis-map coefficients of A112227 → (θ+1)² → basis A013611
- basis-map coefficients of A021012 → θ+1 → basis A013611
- basis-map coefficients of A112227 → basis A013611 → reverse
- A013611 → D → D
- A027467 → basis A013611
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
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.