basis-map coefficients of A030528, shifted → basis A013609 → (θ+1)²
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-1 - 6\,x - 12\,x^{2} - 8\,x^{3} - 6\,n x - 20\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | -1 | -8 | -9 | ||||||
| 2 | 1 | 16 | 36 | 53 | |||||
| 3 | -1 | -24 | -108 | -128 | -261 | ||||
| 4 | 1 | 32 | 216 | 512 | 400 | 1161 | |||
| 5 | -1 | -40 | -360 | -1280 | -2000 | -1152 | -4833 | ||
| 6 | 1 | 48 | 540 | 2560 | 6000 | 6912 | 3136 | 19197 | |
| 7 | -1 | -56 | -756 | -4480 | -14000 | -24192 | -21952 | -8192 | -73629 |
Compute more
?
Technical details
- Node id
n-82c0885ccc3f590d· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-181bf1738e48841f
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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: -1, -8; 2: 1, 16, 36; 3: -1, -24, -108, -128; 4: 1, 32, 216, 512, 400; 5: -1, -40, -360, -1280, -2000, -1152; 6: 1, 48, 540, 2560, 6000, 6912, 3136; 7: -1, -56, -756, -4480, -14000, -24192, -21952, -8192; 8: 1, 64, 1008, 7168, 28000, 64512, 87808, 65536, 20736; 9: -1, -72, -1296, -10752, -50400, -145152, -263424, -294912, -186624, -51200;
%F FormulaThe row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = -(1+(6*n+6)*x+(4*n^2+20*n+12)*x^2+(8*n^2+16*n+8)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + (6*n + 6)*x + (4*n^2 + 20*n + 12)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*x + 4*n^2*x^2)]];
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] = (-(1 + (6*n + 6)*x + (4*n^2 + 20*n + 12)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(1 + (6*n - 2)*x + 4*n^2*x^2)); 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 (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
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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
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
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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from
basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
Related ?
Same fingerprint ?
- A178821 → basis A013611 → (θ+1)²
- A178820 → basis A013611 → (θ+1)²
- A028326 → basis A013611 → (θ+1)²
- A156992 → basis A013611 → (θ+1)²
- A085880 → basis A013611 → (θ+1)²
- A136572 → basis A013609 → (θ+1)²
- basis-map coefficients of A026729 → basis A013609 → (θ+1)²
- basis-map coefficients of A026729, shifted → basis A013609 → (θ+1)²
- basis-map coefficients of A112227 → basis A013609 → (θ+1)²
- A130123 → basis A013609 → (θ+1)²
- A196347 → basis A013611 → (θ+1)²
- A013609 → (θ+1)²
- A178822 → basis A013611 → (θ+1)²
- A120909 → basis A013610 → (θ+1)²
- A085881 → basis A013611 → (θ+1)²
- A038208 → basis A013611 → (θ+1)²
- A120909 → (θ+1)²
- A013609 → D → (θ+1)²
- A013609 → basis A013610 → (θ+1)²
- A356546 → basis A013611 → (θ+1)²
- A038221 → basis A013611 → (θ+1)²
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.