A142461 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(25 - 60\,n + 36\,n^{2}\right)\,P_{n}(x) = \left(-25 - 8140\,x + 825\,x^{2} - 120\,n + 3384\,n x - 17280\,n x^{2} + 180\,n^{2} + 2160\,n^{2} x + 46008\,n^{2} x^{2} - 864\,n^{3} x - 41472\,n^{3} x^{2} + 11664\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(1320\,x - 7100\,x^{2} + 1300\,x^{3} + 288\,n x - 1800\,n x^{2} - 21720\,n x^{3} - 432\,n^{2} x + 3888\,n^{2} x^{2} + 50112\,n^{2} x^{3} + 864\,n^{3} x^{2} - 35424\,n^{3} x^{3} + 5184\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(1260\,x^{2} - 1560\,x^{3} + 300\,x^{4} + 864\,n x^{2} + 3456\,n x^{3} - 4320\,n x^{4} - 1296\,n^{2} x^{2} - 7776\,n^{2} x^{3} + 9072\,n^{2} x^{4} + 5184\,n^{3} x^{3} - 5184\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(50 - 500\,x + 60\,n + 1800\,n x - 144\,n^{2} - 2160\,n^{2} x + 864\,n^{3} x\right)\,P_{n-1}(x) + \left(-60\,x - 100\,x^{2} - 504\,n x - 360\,n x^{2} + 432\,n^{2} x + 1296\,n^{2} x^{2} - 864\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 1 | 56 | 9 | 66 | |||||
| 3 | 1 | 444 | 999 | 16 | 1460 | ||||
| 4 | 1 | 3184 | 25974 | 12736 | 25 | 41920 | |||
| 5 | 1 | 22388 | 473778 | 842272 | 139925 | 36 | 1478400 | ||
| 6 | 1 | 156840 | 7418439 | 32006336 | 20606775 | 1411560 | 49 | 61600000 | |
| 7 | 1 | 1098028 | 107379297 | 937828784 | 1465357475 | 429517188 | 13450843 | 64 | 2954631680 |
Compute more
?
Technical details
- Node id
n-67b2c6109797c507· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-bade81768928a835
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: 1, 56, 9; 3: 1, 444, 999, 16; 4: 1, 3184, 25974, 12736, 25; 5: 1, 22388, 473778, 842272, 139925, 36; 6: 1, 156840, 7418439, 32006336, 20606775, 1411560, 49; 7: 1, 1098028, 107379297, 937828784, 1465357475, 429517188, 13450843, 64; 8: 1, 7686368, 1487341692, 23736539264, 73267873750, 53407213344, 8097749212, 122981888, 81; 9: 1, 53804772, 20079098100, 548828277312, 2981435464350, 4293267068664, 1680786599268, 142784697600, 1089546633, 100;
%F FormulaThe row polynomials satisfy (36*n^2-60*n+25)*P(n,x) = (180*n^2-120*n-25-(864*n^3-2160*n^2-3384*n+8140)*x+(11664*n^4-41472*n^3+46008*n^2-17280*n+825)*x^2)*P(n-2,x) - ((432*n^2-288*n-1320)*x-(864*n^3+3888*n^2-1800*n-7100)*x^2-(5184*n^4-35424*n^3+50112*n^2-21720*n+1300)*x^3)*d/dx P(n-2,x) - ((1296*n^2-864*n-1260)*x^2-(5184*n^3-7776*n^2+3456*n-1560)*x^3+(5184*n^3-9072*n^2+4320*n-300)*x^4)*d^2/dx^2 P(n-2,x) - (144*n^2-60*n-50-(864*n^3-2160*n^2+1800*n-500)*x)*P(n-1,x) + ((432*n^2-504*n-60)*x-(864*n^3-1296*n^2+360*n+100)*x^2)*d/dx 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[((180*n^2 - 120*n - 25 - (864*n^3 - 2160*n^2 - 3384*n + 8140)*x + (11664*n^4 - 41472*n^3 + 46008*n^2 - 17280*n + 825)*x^2)*P[n - 2, x] - ((432*n^2 - 288*n - 1320)*x - (864*n^3 + 3888*n^2 - 1800*n - 7100)*x^2 - (5184*n^4 - 35424*n^3 + 50112*n^2 - 21720*n + 1300)*x^3)*D[P[n - 2, x], x] - ((1296*n^2 - 864*n - 1260)*x^2 - (5184*n^3 - 7776*n^2 + 3456*n - 1560)*x^3 + (5184*n^3 - 9072*n^2 + 4320*n - 300)*x^4)*D[P[n - 2, x], {x, 2}] - (144*n^2 - 60*n - 50 - (864*n^3 - 2160*n^2 + 1800*n - 500)*x)*P[n - 1, x] + ((432*n^2 - 504*n - 60)*x - (864*n^3 - 1296*n^2 + 360*n + 100)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 60*n + 25)]];
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] = ((180*n^2 - 120*n - 25 - (864*n^3 - 2160*n^2 - 3384*n + 8140)*x + (11664*n^4 - 41472*n^3 + 46008*n^2 - 17280*n + 825)*x^2)*v[i - 2] - ((432*n^2 - 288*n - 1320)*x - (864*n^3 + 3888*n^2 - 1800*n - 7100)*x^2 - (5184*n^4 - 35424*n^3 + 50112*n^2 - 21720*n + 1300)*x^3)*deriv(v[i - 2], x) - ((1296*n^2 - 864*n - 1260)*x^2 - (5184*n^3 - 7776*n^2 + 3456*n - 1560)*x^3 + (5184*n^3 - 9072*n^2 + 4320*n - 300)*x^4)*deriv(deriv(v[i - 2], x), x) - (144*n^2 - 60*n - 50 - (864*n^3 - 2160*n^2 + 1800*n - 500)*x)*v[i - 1] + ((432*n^2 - 504*n - 60)*x - (864*n^3 - 1296*n^2 + 360*n + 100)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 60*n + 25)); 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]
-
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 ›
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.