basis-map coefficients of A127647 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - 5\,x + 5\,x^{2} - x^{3} + 6\,n x - 17\,n x^{2} + 5\,n x^{3} + 11\,n^{2} x^{2} - 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} - 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,x^{2} - 2\,x^{3} - 3\,x^{4} - x^{5} + 6\,n x + 17\,n x^{2} + 17\,n x^{3} + 7\,n x^{4} + n x^{5} + 11\,n^{2} x^{2} + 29\,n^{2} x^{3} + 26\,n^{2} x^{4} + 9\,n^{2} x^{5} + n^{2} x^{6} + 6\,n^{3} x^{3} + 14\,n^{3} x^{4} + 10\,n^{3} x^{5} + 2\,n^{3} x^{6} + n^{4} x^{4} + 2\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-2}(x) + \left(1 - x - 8\,x^{2} - 8\,x^{3} - x^{4} + x^{5} + 6\,n x - 12\,n x^{3} - 6\,n x^{4} + 11\,n^{2} x^{2} + 9\,n^{2} x^{3} - 4\,n^{2} x^{4} - 2\,n^{2} x^{5} + 6\,n^{3} x^{3} + 6\,n^{3} x^{4} + n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 2 | 16 | 18 | 36 | |||||
| 3 | 3 | 36 | 81 | 48 | 168 | ||||
| 4 | 5 | 80 | 270 | 320 | 125 | 800 | |||
| 5 | 8 | 160 | 720 | 1280 | 1000 | 288 | 3456 | ||
| 6 | 13 | 312 | 1755 | 4160 | 4875 | 2808 | 637 | 14560 | |
| 7 | 21 | 588 | 3969 | 11760 | 18375 | 15876 | 7203 | 1344 | 59136 |
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Technical details
- Node id
n-afcabc33edb2f88e· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-d42e6339eab21b9c
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): yes - 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, 4; 2: 2, 16, 18; 3: 3, 36, 81, 48; 4: 5, 80, 270, 320, 125; 5: 8, 160, 720, 1280, 1000, 288; 6: 13, 312, 1755, 4160, 4875, 2808, 637; 7: 21, 588, 3969, 11760, 18375, 15876, 7203, 1344; 8: 34, 1088, 8568, 30464, 59500, 68544, 46648, 17408, 2754; 9: 55, 1980, 17820, 73920, 173250, 249480, 226380, 126720, 40095, 5500;
%F FormulaThe row polynomials satisfy (1+(6*n-5)*x+(11*n^2-17*n+5)*x^2+(6*n^3-11*n^2+5*n-1)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = (1+(6*n+3)*x+(11*n^2+17*n+2)*x^2+(6*n^3+29*n^2+17*n-2)*x^3+(n^4+14*n^3+26*n^2+7*n-3)*x^4+(2*n^4+10*n^3+9*n^2+n-1)*x^5+(n^4+2*n^3+n^2)*x^6)*P(n-2,x) + (1+(6*n-1)*x+(11*n^2-8)*x^2+(6*n^3+9*n^2-12*n-8)*x^3+(n^4+6*n^3-4*n^2-6*n-1)*x^4+(n^4-2*n^2+1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (6*n + 3)*x + (11*n^2 + 17*n + 2)*x^2 + (6*n^3 + 29*n^2 + 17*n - 2)*x^3 + (n^4 + 14*n^3 + 26*n^2 + 7*n - 3)*x^4 + (2*n^4 + 10*n^3 + 9*n^2 + n - 1)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*P[n - 2, x] + (1 + (6*n - 1)*x + (11*n^2 - 8)*x^2 + (6*n^3 + 9*n^2 - 12*n - 8)*x^3 + (n^4 + 6*n^3 - 4*n^2 - 6*n - 1)*x^4 + (n^4 - 2*n^2 + 1)*x^5)*P[n - 1, x])/(1 + (6*n - 5)*x + (11*n^2 - 17*n + 5)*x^2 + (6*n^3 - 11*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((1 + (6*n + 3)*x + (11*n^2 + 17*n + 2)*x^2 + (6*n^3 + 29*n^2 + 17*n - 2)*x^3 + (n^4 + 14*n^3 + 26*n^2 + 7*n - 3)*x^4 + (2*n^4 + 10*n^3 + 9*n^2 + n - 1)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*v[i - 2] + (1 + (6*n - 1)*x + (11*n^2 - 8)*x^2 + (6*n^3 + 9*n^2 - 12*n - 8)*x^3 + (n^4 + 6*n^3 - 4*n^2 - 6*n - 1)*x^4 + (n^4 - 2*n^2 + 1)*x^5)*v[i - 1])/(1 + (6*n - 5)*x + (11*n^2 - 17*n + 5)*x^2 + (6*n^3 - 11*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)); 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 coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).?
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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
basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).?
Related ?
Same fingerprint ?
- A038325 → (θ+1)²
- A135838 → (θ+1)²
- A156992 → (θ+1)²
- basis-map coefficients of A113214, shifted → D → (θ+1)²
- A178820 → basis A013609 → (θ+1)²
- basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
- A178821 → basis A013609 → (θ+1)²
- basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
- A028326 → basis A013609 → (θ+1)²
- A038208 → D → (θ+1)²
- A085881 → basis A013609 → (θ+1)²
- A028326 → (θ+1)²
- basis-map coefficients of A026729 → basis A007318 → (θ+1)²
- A085880 → D → (θ+1)²
- basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
- A134346 → (θ+1)²
- A156992 → basis A013609 → (θ+1)²
- A125092 → reverse → D
- A178822 → (θ+1)²
- A003506 → (θ+1)²
- A028326 → D → (θ+1)²
- A038330 → basis A013611 → (θ+1)²
- A085881 → (θ+1)²
- A038325 → basis A013609 → (θ+1)²
- A131431 → x ↦ x+1 → (θ+1)²
- A178821 → (θ+1)²
- A196347 → (θ+1)²
- A135065 → (θ+1)²
- A356546 → basis A013609 → (θ+1)²
- A085880 → (θ+1)²
- A178820 → (θ+1)²
- A344565 → (θ+1)²
- A003506 → D → (θ+1)²
- basis-map coefficients of A113214, shifted → (θ+1)²
- A178822 → basis A013609 → (θ+1)²
- A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
- A038221 → (θ+1)²
- A038338 → basis A013609 → (θ+1)²
- A125092 → reverse
- A196347 → basis A013609 → (θ+1)²
- A124860 → (θ+1)²
- A085880 → basis A013609 → (θ+1)²
- A038208 → (θ+1)²
- A038338 → (θ+1)²
- A038221 → basis A013609 → (θ+1)²
- A376827 → (θ+1)²
- A120910 → basis A013610 → (θ+1)²
- A027465 → basis A013611 → (θ+1)²
- A356546 → (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.