A290315 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(81 - 72\,n + 16\,n^{2}\right)\,P_{n}(x) = \left(45 + 3256\,x + 174636\,x^{2} - 56\,n - 6112\,n x - 347040\,n x^{2} + 16\,n^{2} + 3200\,n^{2} x + 251648\,n^{2} x^{2} - 512\,n^{3} x - 79360\,n^{3} x^{2} + 9216\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-156\,x - 2248\,x^{2} + 124416\,x^{3} + 224\,n x + 3808\,n x^{2} - 229824\,n x^{3} - 64\,n^{2} x - 2432\,n^{2} x^{2} + 151296\,n^{2} x^{3} + 512\,n^{3} x^{2} - 41984\,n^{3} x^{3} + 4096\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-164\,x^{2} - 6800\,x^{3} + 14256\,x^{4} + 224\,n x^{2} + 11072\,n x^{3} - 23040\,n x^{4} - 64\,n^{2} x^{2} - 5888\,n^{2} x^{3} + 12032\,n^{2} x^{4} + 1024\,n^{3} x^{3} - 2048\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(36 - 4536\,x - 16\,n + 6624\,n x - 3200\,n^{2} x + 512\,n^{3} x\right)\,P_{n-1}(x) + \left(244\,x + 3240\,x^{2} - 256\,n x - 5472\,n x^{2} + 64\,n^{2} x + 2944\,n^{2} x^{2} - 512\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 8 | 9 | ||||||
| 3 | 1 | 64 | 108 | 173 | |||||
| 4 | 1 | 264 | 2556 | 1920 | 4741 | ||||
| 5 | 1 | 896 | 25848 | 95232 | 42000 | 163977 | |||
| 6 | 1 | 2824 | 189720 | 1861376 | 3651600 | 1088640 | 6794161 | ||
| 7 | 1 | 8640 | 1190052 | 24497152 | 120116400 | 148690944 | 32598720 | 327101909 | |
| 8 | 1 | 26120 | 6844500 | 260337280 | 2535566000 | 7516499328 | 6495330240 | 1107025920 | 17921629389 |
Compute more
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Technical details
- Node id
n-6f6f9db0a6c695a0· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-4ace4b2db4280241
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: 1, 8; 3: 1, 64, 108; 4: 1, 264, 2556, 1920; 5: 1, 896, 25848, 95232, 42000; 6: 1, 2824, 189720, 1861376, 3651600, 1088640; 7: 1, 8640, 1190052, 24497152, 120116400, 148690944, 32598720; 8: 1, 26120, 6844500, 260337280, 2535566000, 7516499328, 6495330240, 1107025920; 9: 1, 78592, 37396080, 2430760960, 41531836000, 236032524288, 473209341696, 305051074560, 42032390400; 10: 1, 236040, 197943984, 20871278080, 578881980000, 5586649959168, 20929980046080, 30534678577152, 15385600753920, 1764322560000;
%F FormulaThe row polynomials satisfy (16*n^2-72*n+81)*P(n,x) = (16*n^2-56*n+45-(512*n^3-3200*n^2+6112*n-3256)*x+(9216*n^4-79360*n^3+251648*n^2-347040*n+174636)*x^2)*P(n-2,x) - ((64*n^2-224*n+156)*x-(512*n^3-2432*n^2+3808*n-2248)*x^2-(4096*n^4-41984*n^3+151296*n^2-229824*n+124416)*x^3)*d/dx P(n-2,x) - ((64*n^2-224*n+164)*x^2-(1024*n^3-5888*n^2+11072*n-6800)*x^3+(2048*n^3-12032*n^2+23040*n-14256)*x^4)*d^2/dx^2 P(n-2,x) - (16*n-36-(512*n^3-3200*n^2+6624*n-4536)*x)*P(n-1,x) + ((64*n^2-256*n+244)*x-(512*n^3-2944*n^2+5472*n-3240)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 56*n + 45 - (512*n^3 - 3200*n^2 + 6112*n - 3256)*x + (9216*n^4 - 79360*n^3 + 251648*n^2 - 347040*n + 174636)*x^2)*P[n - 2, x] - ((64*n^2 - 224*n + 156)*x - (512*n^3 - 2432*n^2 + 3808*n - 2248)*x^2 - (4096*n^4 - 41984*n^3 + 151296*n^2 - 229824*n + 124416)*x^3)*D[P[n - 2, x], x] - ((64*n^2 - 224*n + 164)*x^2 - (1024*n^3 - 5888*n^2 + 11072*n - 6800)*x^3 + (2048*n^3 - 12032*n^2 + 23040*n - 14256)*x^4)*D[P[n - 2, x], {x, 2}] - (16*n - 36 - (512*n^3 - 3200*n^2 + 6624*n - 4536)*x)*P[n - 1, x] + ((64*n^2 - 256*n + 244)*x - (512*n^3 - 2944*n^2 + 5472*n - 3240)*x^2)*D[P[n - 1, x], x])/(16*n^2 - 72*n + 81)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = ((16*n^2 - 56*n + 45 - (512*n^3 - 3200*n^2 + 6112*n - 3256)*x + (9216*n^4 - 79360*n^3 + 251648*n^2 - 347040*n + 174636)*x^2)*v[i - 2] - ((64*n^2 - 224*n + 156)*x - (512*n^3 - 2432*n^2 + 3808*n - 2248)*x^2 - (4096*n^4 - 41984*n^3 + 151296*n^2 - 229824*n + 124416)*x^3)*deriv(v[i - 2], x) - ((64*n^2 - 224*n + 164)*x^2 - (1024*n^3 - 5888*n^2 + 11072*n - 6800)*x^3 + (2048*n^3 - 12032*n^2 + 23040*n - 14256)*x^4)*deriv(deriv(v[i - 2], x), x) - (16*n - 36 - (512*n^3 - 3200*n^2 + 6624*n - 4536)*x)*v[i - 1] + ((64*n^2 - 256*n + 244)*x - (512*n^3 - 2944*n^2 + 5472*n - 3240)*x^2)*deriv(v[i - 1], x))/(16*n^2 - 72*n + 81)); 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]
-
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
-
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
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.