basis-map coefficients of A026729, shifted → basis A013610 → (θ+1)²
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - 12\,x + 9\,x^{2} + 9\,n x - 18\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-1 + 9\,x^{2} - 9\,n x - 27\,n x^{2} - 9\,n^{2} x^{2} - 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | -1 | -12 | -13 | ||||||
| 3 | 1 | 24 | 81 | 106 | |||||
| 4 | -1 | -36 | -243 | -432 | -712 | ||||
| 5 | 1 | 48 | 486 | 1728 | 2025 | 4288 | |||
| 6 | -1 | -60 | -810 | -4320 | -10125 | -8748 | -24064 | ||
| 7 | 1 | 72 | 1215 | 8640 | 30375 | 52488 | 35721 | 128512 | |
| 8 | -1 | -84 | -1701 | -15120 | -70875 | -183708 | -250047 | -139968 | -661504 |
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Technical details
- Node id
n-bc8fa0dba1857955· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-10c18c8945c6e24a
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): 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: -1, -12; 3: 1, 24, 81; 4: -1, -36, -243, -432; 5: 1, 48, 486, 1728, 2025; 6: -1, -60, -810, -4320, -10125, -8748; 7: 1, 72, 1215, 8640, 30375, 52488, 35721; 8: -1, -84, -1701, -15120, -70875, -183708, -250047, -139968; 9: 1, 96, 2268, 24192, 141750, 489888, 1000188, 1119744, 531441; 10: -1, -108, -2916, -36288, -255150, -1102248, -3000564, -5038848, -4782969, -1968300;
%F FormulaThe row polynomials satisfy (1+(9*n-12)*x+(9*n^2-18*n+9)*x^2)*P(n,x) = -(1+9*n*x+(9*n^2+27*n-9)*x^2+27*n^2*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + 9*n*x + (9*n^2 + 27*n - 9)*x^2 + 27*n^2*x^3)*P[n - 1, x])/(1 + (9*n - 12)*x + (9*n^2 - 18*n + 9)*x^2)]];
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] = (-(1 + 9*n*x + (9*n^2 + 27*n - 9)*x^2 + 27*n^2*x^3)*v[i - 1])/(1 + (9*n - 12)*x + (9*n^2 - 18*n + 9)*x^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 multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, 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_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
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 ?
- A013610 → D → (θ+1)²
- A013610 → basis A013611 → (θ+1)²
- basis-map coefficients of A112227 → basis A013610 → (θ+1)²
- A130123 → basis A013610 → (θ+1)²
- basis-map coefficients of A030528, shifted → basis A013610 → (θ+1)²
- A136572 → basis A013610 → (θ+1)²
- A013610 → (θ+1)²
- basis-map coefficients of A026729 → basis A013610 → (θ+1)²
References
- [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.