A167884 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(49 - 112\,n + 64\,n^{2}\right)\,P_{n}(x) = \left(-49 - 27300\,x + 5145\,x^{2} - 336\,n + 11680\,n x - 67424\,n x^{2} + 448\,n^{2} + 5376\,n^{2} x + 162176\,n^{2} x^{2} - 2048\,n^{3} x - 137216\,n^{3} x^{2} + 36864\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4144\,x - 23100\,x^{2} + 7252\,x^{3} + 768\,n x - 7072\,n x^{2} - 82432\,n x^{3} - 1024\,n^{2} x + 13056\,n^{2} x^{2} + 172544\,n^{2} x^{3} + 2048\,n^{3} x^{2} - 114688\,n^{3} x^{3} + 16384\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4032\,x^{2} - 5600\,x^{3} + 1568\,x^{4} + 3072\,n x^{2} + 13056\,n x^{3} - 16128\,n x^{4} - 4096\,n^{2} x^{2} - 26624\,n^{2} x^{3} + 30720\,n^{2} x^{4} + 16384\,n^{3} x^{3} - 16384\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(98 - 1372\,x + 224\,n + 4704\,n x - 384\,n^{2} - 5376\,n^{2} x + 2048\,n^{3} x\right)\,P_{n-1}(x) + \left(-112\,x - 196\,x^{2} - 1280\,n x - 1120\,n x^{2} + 1024\,n^{2} x + 3328\,n^{2} x^{2} - 2048\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 1 | 72 | 9 | 82 | |||||
| 3 | 1 | 716 | 1611 | 16 | 2344 | ||||
| 4 | 1 | 6544 | 54774 | 26176 | 25 | 87520 | |||
| 5 | 1 | 59028 | 1299258 | 2309792 | 368925 | 36 | 4037040 | ||
| 6 | 1 | 531416 | 26470215 | 115489600 | 73528375 | 4782744 | 49 | 220802400 | |
| 7 | 1 | 4782940 | 499016781 | 4440159280 | 6937748875 | 1996067124 | 58591015 | 64 | 13936366080 |
Compute more
?
Technical details
- Node id
n-622e3f01c07ad062· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-1796614eedcc04be
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, 72, 9; 3: 1, 716, 1611, 16; 4: 1, 6544, 54774, 26176, 25; 5: 1, 59028, 1299258, 2309792, 368925, 36; 6: 1, 531416, 26470215, 115489600, 73528375, 4782744, 49; 7: 1, 4782940, 499016781, 4440159280, 6937748875, 1996067124, 58591015, 64; 8: 1, 43046688, 9010604412, 147376760704, 457891425750, 331597711584, 49057735132, 688747008, 81; 9: 1, 387420452, 158701012740, 4469342779712, 24551740782350, 35354506726584, 13687362262868, 1128540535040, 7845264153, 100;
%F FormulaThe row polynomials satisfy (64*n^2-112*n+49)*P(n,x) = (448*n^2-336*n-49-(2048*n^3-5376*n^2-11680*n+27300)*x+(36864*n^4-137216*n^3+162176*n^2-67424*n+5145)*x^2)*P(n-2,x) - ((1024*n^2-768*n-4144)*x-(2048*n^3+13056*n^2-7072*n-23100)*x^2-(16384*n^4-114688*n^3+172544*n^2-82432*n+7252)*x^3)*d/dx P(n-2,x) - ((4096*n^2-3072*n-4032)*x^2-(16384*n^3-26624*n^2+13056*n-5600)*x^3+(16384*n^3-30720*n^2+16128*n-1568)*x^4)*d^2/dx^2 P(n-2,x) - (384*n^2-224*n-98-(2048*n^3-5376*n^2+4704*n-1372)*x)*P(n-1,x) + ((1024*n^2-1280*n-112)*x-(2048*n^3-3328*n^2+1120*n+196)*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[((448*n^2 - 336*n - 49 - (2048*n^3 - 5376*n^2 - 11680*n + 27300)*x + (36864*n^4 - 137216*n^3 + 162176*n^2 - 67424*n + 5145)*x^2)*P[n - 2, x] - ((1024*n^2 - 768*n - 4144)*x - (2048*n^3 + 13056*n^2 - 7072*n - 23100)*x^2 - (16384*n^4 - 114688*n^3 + 172544*n^2 - 82432*n + 7252)*x^3)*D[P[n - 2, x], x] - ((4096*n^2 - 3072*n - 4032)*x^2 - (16384*n^3 - 26624*n^2 + 13056*n - 5600)*x^3 + (16384*n^3 - 30720*n^2 + 16128*n - 1568)*x^4)*D[P[n - 2, x], {x, 2}] - (384*n^2 - 224*n - 98 - (2048*n^3 - 5376*n^2 + 4704*n - 1372)*x)*P[n - 1, x] + ((1024*n^2 - 1280*n - 112)*x - (2048*n^3 - 3328*n^2 + 1120*n + 196)*x^2)*D[P[n - 1, x], x])/(64*n^2 - 112*n + 49)]];
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] = ((448*n^2 - 336*n - 49 - (2048*n^3 - 5376*n^2 - 11680*n + 27300)*x + (36864*n^4 - 137216*n^3 + 162176*n^2 - 67424*n + 5145)*x^2)*v[i - 2] - ((1024*n^2 - 768*n - 4144)*x - (2048*n^3 + 13056*n^2 - 7072*n - 23100)*x^2 - (16384*n^4 - 114688*n^3 + 172544*n^2 - 82432*n + 7252)*x^3)*deriv(v[i - 2], x) - ((4096*n^2 - 3072*n - 4032)*x^2 - (16384*n^3 - 26624*n^2 + 13056*n - 5600)*x^3 + (16384*n^3 - 30720*n^2 + 16128*n - 1568)*x^4)*deriv(deriv(v[i - 2], x), x) - (384*n^2 - 224*n - 98 - (2048*n^3 - 5376*n^2 + 4704*n - 1372)*x)*v[i - 1] + ((1024*n^2 - 1280*n - 112)*x - (2048*n^3 - 3328*n^2 + 1120*n + 196)*x^2)*deriv(v[i - 1], x))/(64*n^2 - 112*n + 49)); 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
-
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
-
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.