A376827 → θ+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n - n x + 2\,n^{2} x - n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-3 - 9\,x - 9\,x^{2} - 3\,x^{3} + 3\,n + 3\,n x - 6\,n x^{2} - 9\,n x^{3} - 3\,n x^{4} + 6\,n^{2} x + 12\,n^{2} x^{2} + 6\,n^{2} x^{3} + 3\,n^{3} x^{2} + 6\,n^{3} x^{3} + 3\,n^{3} x^{4}\right)\,P_{n-2}(x) + \left(-1 - x + x^{2} + x^{3} + 2\,n - 4\,n x^{2} - 2\,n x^{3} + 4\,n^{2} x + 3\,n^{2} x^{2} - n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 3 | 12 | 9 | 24 | |||||
| 3 | 7 | 42 | 63 | 28 | 140 | ||||
| 4 | 19 | 152 | 342 | 304 | 95 | 912 | |||
| 5 | 51 | 510 | 1530 | 2040 | 1275 | 306 | 5712 | ||
| 6 | 141 | 1692 | 6345 | 11280 | 10575 | 5076 | 987 | 36096 | |
| 7 | 393 | 5502 | 24759 | 55020 | 68775 | 49518 | 19257 | 3144 | 226368 |
Compute more
?
Technical details
- Node id
n-7a5b2f8a6a89e281· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-e99c3824c4864868
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 2; 2: 3, 12, 9; 3: 7, 42, 63, 28; 4: 19, 152, 342, 304, 95; 5: 51, 510, 1530, 2040, 1275, 306; 6: 141, 1692, 6345, 11280, 10575, 5076, 987; 7: 393, 5502, 24759, 55020, 68775, 49518, 19257, 3144; 8: 1107, 17712, 92988, 247968, 387450, 371952, 216972, 70848, 9963; 9: 3139, 56502, 339012, 1054704, 1977570, 2373084, 1845732, 904032, 254259, 31390;
%F FormulaThe row polynomials satisfy (n+(2*n^2-n)*x+(n^3-n^2)*x^2)*P(n,x) = (3*n-3+(6*n^2+3*n-9)*x+(3*n^3+12*n^2-6*n-9)*x^2+(6*n^3+6*n^2-9*n-3)*x^3+(3*n^3-3*n)*x^4)*P(n-2,x) + (2*n-1+(4*n^2-1)*x+(2*n^3+3*n^2-4*n+1)*x^2+(2*n^3-n^2-2*n+1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (6*n^2 + 3*n - 9)*x + (3*n^3 + 12*n^2 - 6*n - 9)*x^2 + (6*n^3 + 6*n^2 - 9*n - 3)*x^3 + (3*n^3 - 3*n)*x^4)*P[n - 2, x] + (2*n - 1 + (4*n^2 - 1)*x + (2*n^3 + 3*n^2 - 4*n + 1)*x^2 + (2*n^3 - n^2 - 2*n + 1)*x^3)*P[n - 1, x])/(n + (2*n^2 - n)*x + (n^3 - 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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (6*n^2 + 3*n - 9)*x + (3*n^3 + 12*n^2 - 6*n - 9)*x^2 + (6*n^3 + 6*n^2 - 9*n - 3)*x^3 + (3*n^3 - 3*n)*x^4)*v[i - 2] + (2*n - 1 + (4*n^2 - 1)*x + (2*n^3 + 3*n^2 - 4*n + 1)*x^2 + (2*n^3 - n^2 - 2*n + 1)*x^3)*v[i - 1])/(n + (2*n^2 - n)*x + (n^3 - 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
T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
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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
T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
Related ?
Same fingerprint ?
- A356546 → basis A013609 → D+1
- A038338 → basis A013609 → D+1
- A094305 → D+1
- A344565 → θ+1
- A356546 → basis A013609 → θ+1
- A038208 → θ+1
- A085880 → D+1
- A038338 → θ+1
- A156992 → basis A013609 → θ+1
- A178822 → basis A013609 → D+1
- A178820 → basis A013609 → θ+1
- A178820 → basis A013609 → D+1
- A131431 → x ↦ x+1 → θ+1
- A085881 → θ+1
- A038221 → θ+1
- A196347 → basis A013609 → θ+1
- A028326 → D → D+1
- basis-map coefficients of A127647 → θ+1
- A093375 → D → D
- A085880 → basis A013609 → θ+1
- A028326 → basis A013609 → θ+1
- A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
- A028326 → D → θ+1
- A038221 → D+1
- A085880 → basis A013609 → D+1
- A156992 → basis A013609 → D+1
- basis-map coefficients of A113214, shifted → D+1
- basis-map coefficients of A113214, shifted → basis A013609 → θ+1
- A027465 → basis A013611 → D+1
- A178821 → θ+1
- A178821 → D+1
- A135065 → D+1
- A124860 → θ+1
- A134346 → D+1
- basis-map coefficients of A112227 → x ↦ x+1 → D+1
- A356546 → θ+1
- A038338 → D+1
- A038338 → basis A013609 → θ+1
- A120910 → basis A013610 → θ+1
- A085881 → basis A013609 → D+1
- basis-map coefficients of A113214, shifted → D → D+1
- A120910 → basis A013610 → D+1
- A003506 → D → θ+1
- A038325 → basis A013609 → θ+1
- A028326 → basis A013609 → D+1
- A028326 → D+1
- A085880 → D → θ+1
- A094305 → θ+1
- A038325 → D+1
- basis-map coefficients of A127647 → D+1
- basis-map coefficients of A113214, shifted → basis A013609 → D+1
- A085881 → D+1
- A156992 → D+1
- A038221 → basis A013609 → θ+1
- basis-map coefficients of A113214, shifted → θ+1
- A156992 → θ+1
- basis-map coefficients of A026729 → basis A007318 → D+1
- A038330 → basis A013611 → θ+1
- A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
- A134346 → θ+1
- A135838 → D+1
- A027465 → basis A013611 → θ+1
- A178820 → θ+1
- A196347 → D+1
- A344565 → D+1
- A135838 → θ+1
- basis-map coefficients of A026729 → basis A007318 → θ+1
- A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
- A007318 → ×(−(1/2)n+(1/2)n²) → D+1
- A124860 → D+1
- A178822 → basis A013609 → θ+1
- A196347 → basis A013609 → D+1
- basis-map coefficients of A113214, shifted → D → θ+1
- A085881 → basis A013609 → θ+1
- A178822 → D+1
- basis-map coefficients of A026729, shifted → basis A007318 → θ+1
- A196347 → θ+1
- A038208 → D+1
- A178821 → basis A013609 → D+1
- A085880 → D → D+1
- basis-map coefficients of A026729, shifted → basis A007318 → D+1
- A085880 → θ+1
- A038208 → D → θ+1
- A178821 → basis A013609 → θ+1
- A376827 → D+1
- A038221 → basis A013609 → D+1
- A178820 → D+1
- A038325 → θ+1
- A038208 → D → D+1
- A131431 → x ↦ x+1 → D+1
- A356546 → D+1
- A135065 → θ+1
- basis-map coefficients of A112227 → x ↦ x+1 → θ+1
- A216973 → θ+1
- A038330 → basis A013611 → D+1
- A178822 → θ+1
- A131431 → D+1 → basis A007318
- A093375 → D
- A038325 → basis A013609 → D+1
- A028326 → θ+1
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.