basis-map coefficients of A030528, shifted → basis A013610 → reverse
only rational roots
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-3 - x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | -3 | -1 | -4 | |||||||||||||
| 2 | 9 | 6 | 1 | 16 | ||||||||||||
| 3 | -27 | -27 | -9 | -1 | -64 | |||||||||||
| 4 | 81 | 108 | 54 | 12 | 1 | 256 | ||||||||||
| 5 | -243 | -405 | -270 | -90 | -15 | -1 | -1024 | |||||||||
| 6 | 729 | 1458 | 1215 | 540 | 135 | 18 | 1 | 4096 | ||||||||
| 7 | -2187 | -5103 | -5103 | -2835 | -945 | -189 | -21 | -1 | -16384 | |||||||
| 8 | 6561 | 17496 | 20412 | 13608 | 5670 | 1512 | 252 | 24 | 1 | 65536 | ||||||
| 9 | -19683 | -59049 | -78732 | -61236 | -30618 | -10206 | -2268 | -324 | -27 | -1 | -262144 | |||||
| 10 | 59049 | 196830 | 295245 | 262440 | 153090 | 61236 | 17010 | 3240 | 405 | 30 | 1 | 1048576 | ||||
| 11 | -177147 | -649539 | -1082565 | -1082565 | -721710 | -336798 | -112266 | -26730 | -4455 | -495 | -33 | -1 | -4194304 | |||
| 12 | 531441 | 2125764 | 3897234 | 4330260 | 3247695 | 1732104 | 673596 | 192456 | 40095 | 5940 | 594 | 36 | 1 | 16777216 | ||
| 13 | -1594323 | -6908733 | -13817466 | -16888014 | -14073345 | -8444007 | -3752892 | -1250964 | -312741 | -57915 | -7722 | -702 | -39 | -1 | -67108864 | |
| 14 | 4782969 | 22320522 | 48361131 | 64481508 | 59108049 | 39405366 | 19702683 | 7505784 | 2189187 | 486486 | 81081 | 9828 | 819 | 42 | 1 | 268435456 |
Compute more
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Technical details
- Node id
n-3f70946c83a70881· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-f9762973ec1089d7
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 0)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): no - 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: -3, -1; 2: 9, 6, 1; 3: -27, -27, -9, -1; 4: 81, 108, 54, 12, 1; 5: -243, -405, -270, -90, -15, -1; 6: 729, 1458, 1215, 540, 135, 18, 1; 7: -2187, -5103, -5103, -2835, -945, -189, -21, -1; 8: 6561, 17496, 20412, 13608, 5670, 1512, 252, 24, 1; 9: -19683, -59049, -78732, -61236, -30618, -10206, -2268, -324, -27, -1;
%F FormulaThe row polynomials satisfy P(n,x) = -(3+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(3 + x)*P[n - 1, x]];
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] = -(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 - 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)
- Real-rooted? Every row P_n, n ≥ 0, 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
shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
Related ?
Same fingerprint ?
- A038329 → x ↦ x+1 → basis A013609
- A156992 → basis A013611 → x ↦ x/(x+1)
- A013610 → D → D
- A136572 → basis A013610 → D
- A178821 → basis A007318 → x ↦ x+1
- A136572 → basis A013610 → basis A013611
- A130123 → basis A013610 → D
- basis-map coefficients of A112227 → basis A013610 → reverse
- A136572 → D → basis A013610
- A134309 → basis A013610
- image under θ+1 → (θ+1)² → basis A013610
- A013610 → basis A013611 → reverse
- A356546 → x ↦ x+1 → x ↦ x+1
- basis-map coefficients of A026729 → basis A013610
- A178820 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A030528, shifted → θ+1 → basis A013610
- basis-map coefficients of A026729 → (θ+1)² → basis A013610
- A129186 → basis A013610
- A131431 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → D
- basis-map coefficients of A026729, shifted → basis A013610 → reverse
- A013610 → basis A013611 → D
- A131436 → basis A013610
- basis-map coefficients of A112227 → θ+1 → basis A013610
- A028326 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
- A038243 → basis A013609 → D
- A130123 → θ+1 → basis A013610
- basis-map coefficients of A021012 → basis A013610
- A178821 → basis A013611 → x ↦ x/(x+1)
- A038279 → basis A013610
- A131431 → D → basis A013610
- A120910 → basis A007318
- A038243 → basis A013609
- A127648 → basis A013610
- A085880 → x ↦ x+1 → x ↦ x+1
- A027465 → D → D
- A027465 → D
- A038330 → basis A013609 → x ↦ x+1
- A085880 → basis A013611 → x ↦ x/(x+1)
- A028326 → basis A007318 → basis A007318
- A038338 → x ↦ x+1 → basis A007318
- A027465: Cube of lower triangular normalized binomial matrix.
- A196347 → x ↦ x+1 → basis A007318
- basis-map coefficients of A021012 → θ+1 → basis A013610
- A147716 → basis A013610 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013610
- A156992 → basis A007318 → x ↦ x+1
- image under θ+1 → basis A013610
- A038330 → reverse
- A120909 → basis A013610 → x ↦ x/(x+1)
- A130123 → basis A013610
- A013610: Triangle of coefficients in expansion of (1+3*x)^n.
- A038330 → D
- basis-map coefficients of A112227 → basis A013610 → D
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
- A130123 → (θ+1)² → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610
- A136572 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → basis A013611
- basis-map coefficients of A026729 → D → basis A013610
- A132440 → θ+1 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → reverse
- basis-map coefficients of A112227 → (θ+1)² → basis A013610
- A085881 → basis A007318 → basis A007318
- basis-map coefficients of A026729, shifted → basis A013610 → D
- A120910 → basis A007318 → reverse
- A038325 → basis A007318 → basis A007318
- A013610 → D → basis A013611
- A196347 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
- basis-map coefficients of A112227 → basis A013610
- A356546 → basis A013611 → x ↦ x/(x+1)
- A178822 → basis A013611 → x ↦ x/(x+1)
- A094305 → x ↦ x+1 → x ↦ x+1
- A178820 → basis A007318 → basis A007318
- basis-map coefficients of A112227 → basis A013610 → basis A013611
- A178822 → x ↦ x+1 → x ↦ x+1
- A120909 → x ↦ x/(x+1)
- A013609 → basis A013610 → x ↦ x/(x+1)
- A135065 → x ↦ x+1 → basis A007318
- A136521 → basis A013610
- A013610 → D
- A038221 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A030528, shifted → D → basis A013610
- basis-map coefficients of A030528, shifted → basis A013610 → D
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
- basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
- A131431 → θ+1 → basis A013610
- A130123 → basis A013610 → basis A013611
- A120910 → D → x ↦ x+1
- image under θ+1 → θ+1 → D → basis A013610
- A136572 → θ+1 → basis A013610
- A085881 → basis A013611 → x ↦ x/(x+1)
- A127648 → D → basis A013610
- A013610 → basis A013611
- A136572 → (θ+1)² → basis A013610
- A038279 → basis A013610 → D
- A130123 → D → basis A013610
- basis-map coefficients of A026729, shifted → θ+1 → basis A013610
- A038279 → basis A013610 → reverse
- basis-map coefficients of A026729, shifted → D → basis A013610
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.