A144445 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-43 + 42\,n\right)\,P_{n}(x) = \left(49\,x^{2} - 2058\,n x^{2}\right)\,P_{n-4}(x) + \left(-84\,x + 161\,x^{2} + 3528\,n x + 7742\,n x^{2} - 4116\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(49\,x^{2} - 49\,x^{3} - 2058\,n x^{2} + 2058\,n x^{3}\right)\,P_{n-3}'(x) + \left(-6 - 446\,x - 559\,x^{2} + 252\,n - 6048\,n x - 3066\,n x^{2} + 3528\,n^{2} x + 5635\,n^{2} x^{2} - 2058\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-42\,x + 343\,x^{2} - 301\,x^{3} + 1764\,n x + 49\,n x^{2} - 1813\,n x^{3} - 2058\,n^{2} x^{2} + 2058\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-37 + 516\,x - 210\,n - 1106\,n x + 588\,n^{2} x\right)\,P_{n-1}(x) + \left(-301\,x + 301\,x^{2} + 294\,n x - 294\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 46\,x + 3\,x^{2}\)
\(P_{3}(x) = 1 + 412\,x + 618\,x^{2} + 4\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 46 | 3 | 50 | |||||
| 3 | 1 | 412 | 618 | 4 | 1035 | ||||
| 4 | 1 | 3354 | 19023 | 6708 | 5 | 29091 | |||
| 5 | 1 | 26904 | 400353 | 533804 | 67260 | 6 | 1028328 | ||
| 6 | 1 | 215318 | 7210836 | 23664924 | 12018060 | 645954 | 7 | 43755100 | |
| 7 | 1 | 1722644 | 120072204 | 803183948 | 1003979935 | 240144408 | 6029254 | 8 | 2175132402 |
Compute more
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Technical details
- Node id
n-4301181d23f6be2c· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-9e2530a6c09e04bb
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, 2; 2: 1, 46, 3; 3: 1, 412, 618, 4; 4: 1, 3354, 19023, 6708, 5; 5: 1, 26904, 400353, 533804, 67260, 6; 6: 1, 215318, 7210836, 23664924, 12018060, 645954, 7; 7: 1, 1722644, 120072204, 803183948, 1003979935, 240144408, 6029254, 8; 8: 1, 13781266, 1914454437, 23500813784, 58437904315, 35251220676, 4467060353, 55125064, 9; 9: 1, 110250256, 29768499267, 627900631540, 2759263704960, 3311116445952, 1098826105195, 79382664712, 496126152, 10;
%F FormulaThe row polynomials satisfy (42*n-43)*P(n,x) = -(2058*n-49)*x^2*P(n-4,x) + ((3528*n-84)*x-(4116*n^2-7742*n-161)*x^2)*P(n-3,x) - ((2058*n-49)*x^2-(2058*n-49)*x^3)*d/dx P(n-3,x) + (252*n-6+(3528*n^2-6048*n-446)*x-(2058*n^3-5635*n^2+3066*n+559)*x^2)*P(n-2,x) + ((1764*n-42)*x-(2058*n^2-49*n-343)*x^2+(2058*n^2-1813*n-301)*x^3)*d/dx P(n-2,x) - (210*n+37-(588*n^2-1106*n+516)*x)*P(n-1,x) + ((294*n-301)*x-(294*n-301)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 46*x + 3*x^2, P(3,x) = 1 + 412*x + 618*x^2 + 4*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 46*x + 3*x^2;
P[3, x_] := 1 + 412*x + 618*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2058*n - 49)*x^2*P[n - 4, x] + ((3528*n - 84)*x - (4116*n^2 - 7742*n - 161)*x^2)*P[n - 3, x] - ((2058*n - 49)*x^2 - (2058*n - 49)*x^3)*D[P[n - 3, x], x] + (252*n - 6 + (3528*n^2 - 6048*n - 446)*x - (2058*n^3 - 5635*n^2 + 3066*n + 559)*x^2)*P[n - 2, x] + ((1764*n - 42)*x - (2058*n^2 - 49*n - 343)*x^2 + (2058*n^2 - 1813*n - 301)*x^3)*D[P[n - 2, x], x] - (210*n + 37 - (588*n^2 - 1106*n + 516)*x)*P[n - 1, x] + ((294*n - 301)*x - (294*n - 301)*x^2)*D[P[n - 1, x], x])/(42*n - 43)]];
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; v[3] = 1 + 46*x + 3*x^2; v[4] = 1 + 412*x + 618*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(2058*n - 49)*x^2*v[i - 4] + ((3528*n - 84)*x - (4116*n^2 - 7742*n - 161)*x^2)*v[i - 3] - ((2058*n - 49)*x^2 - (2058*n - 49)*x^3)*deriv(v[i - 3], x) + (252*n - 6 + (3528*n^2 - 6048*n - 446)*x - (2058*n^3 - 5635*n^2 + 3066*n + 559)*x^2)*v[i - 2] + ((1764*n - 42)*x - (2058*n^2 - 49*n - 343)*x^2 + (2058*n^2 - 1813*n - 301)*x^3)*deriv(v[i - 2], x) - (210*n + 37 - (588*n^2 - 1106*n + 516)*x)*v[i - 1] + ((294*n - 301)*x - (294*n - 301)*x^2)*deriv(v[i - 1], x))/(42*n - 43)); 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
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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
Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 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
- [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.