A062145 → reverse → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-8 - 18\,x - 2\,x^{2} + 4\,n + 37\,n x + 9\,n x^{2} + 4\,n^{2} - 15\,n^{2} x - 15\,n^{2} x^{2} - 9\,n^{3} x + 10\,n^{3} x^{2} + 5\,n^{4} x - 3\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(16 - 28\,x + 20\,x^{2} + 16\,n - 52\,n x + 12\,n x^{2} - 8\,n x^{3} + 4\,n^{2} + 21\,n^{2} x + 13\,n^{2} x^{2} + 12\,n^{2} x^{3} + 6\,n^{3} x - 52\,n^{3} x^{2} + 5\,n^{4} x + 18\,n^{4} x^{2} - n^{4} x^{3} - 6\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(4\,x - 12\,x^{2} + 8\,x^{3} + 8\,n x - 8\,n x^{2} + 4\,n x^{3} - 4\,n x^{4} - 8\,n^{2} x^{2} + 3\,n^{2} x^{3} + 5\,n^{2} x^{4} + 10\,n^{3} x^{2} - 10\,n^{3} x^{3} - n^{4} x^{3} + n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 4 | 4 | 8 | ||||||
| 3 | 10 | 40 | 9 | 59 | |||||
| 4 | 20 | 180 | 162 | 16 | 378 | ||||
| 5 | 35 | 560 | 1134 | 448 | 25 | 2202 | |||
| 6 | 56 | 1400 | 5040 | 4480 | 1000 | 36 | 12012 | ||
| 7 | 84 | 3024 | 17010 | 26880 | 13500 | 1944 | 49 | 62491 | |
| 8 | 120 | 5880 | 47628 | 117600 | 105000 | 34020 | 3430 | 64 | 313742 |
Compute more
?
Technical details
- Node id
n-6273849bd8391cc0· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-1442d01d433060a6
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): 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 4, 4; 3: 10, 40, 9; 4: 20, 180, 162, 16; 5: 35, 560, 1134, 448, 25; 6: 56, 1400, 5040, 4480, 1000, 36; 7: 84, 3024, 17010, 26880, 13500, 1944, 49; 8: 120, 5880, 47628, 117600, 105000, 34020, 3430, 64; 9: 165, 10560, 116424, 413952, 577500, 332640, 75460, 5632, 81; 10: 220, 17820, 256608, 1241856, 2494800, 2245320, 905520, 152064, 8748, 100;
%F FormulaThe row polynomials satisfy (4*n^2+4*n-8+(5*n^4-9*n^3-15*n^2+37*n-18)*x+(n^6-3*n^5+10*n^3-15*n^2+9*n-2)*x^2)*P(n,x) = (4*n^2+16*n+16+(5*n^4+6*n^3+21*n^2-52*n-28)*x+(n^6+18*n^4-52*n^3+13*n^2+12*n+20)*x^2+(3*n^6-6*n^5-n^4+12*n^2-8*n)*x^3)*P(n-1,x) + ((8*n+4)*x+(10*n^3-8*n^2-8*n-12)*x^2+(2*n^5-n^4-10*n^3+3*n^2+4*n+8)*x^3-(2*n^5-n^4-5*n^2+4*n)*x^4)*d/dx 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[((4*n^2 + 16*n + 16 + (5*n^4 + 6*n^3 + 21*n^2 - 52*n - 28)*x + (n^6 + 18*n^4 - 52*n^3 + 13*n^2 + 12*n + 20)*x^2 + (3*n^6 - 6*n^5 - n^4 + 12*n^2 - 8*n)*x^3)*P[n - 1, x] + ((8*n + 4)*x + (10*n^3 - 8*n^2 - 8*n - 12)*x^2 + (2*n^5 - n^4 - 10*n^3 + 3*n^2 + 4*n + 8)*x^3 - (2*n^5 - n^4 - 5*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 4*n - 8 + (5*n^4 - 9*n^3 - 15*n^2 + 37*n - 18)*x + (n^6 - 3*n^5 + 10*n^3 - 15*n^2 + 9*n - 2)*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] = ((4*n^2 + 16*n + 16 + (5*n^4 + 6*n^3 + 21*n^2 - 52*n - 28)*x + (n^6 + 18*n^4 - 52*n^3 + 13*n^2 + 12*n + 20)*x^2 + (3*n^6 - 6*n^5 - n^4 + 12*n^2 - 8*n)*x^3)*v[i - 1] + ((8*n + 4)*x + (10*n^3 - 8*n^2 - 8*n - 12)*x^2 + (2*n^5 - n^4 - 10*n^3 + 3*n^2 + 4*n + 8)*x^3 - (2*n^5 - n^4 - 5*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 4*n - 8 + (5*n^4 - 9*n^3 - 15*n^2 + 37*n - 18)*x + (n^6 - 3*n^5 + 10*n^3 - 15*n^2 + 9*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, 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
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
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
reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n -…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
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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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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n -…?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.