A257606 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 + 2\,x + n\right)\,P_{n}(x) = \left(-40 - 48\,x - 16\,x^{2} - 12\,x^{3} - 4\,x^{4} - 20\,n - 20\,n x - 24\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} - 8\,n^{2} x - 4\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x - 2\,x^{2} + 6\,x^{3} + 2\,x^{4} + 2\,x^{5} - 4\,n x + n x^{2} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(13 + 21\,x + 6\,x^{2} + 2\,x^{3} + 9\,n + 5\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 8 | 4 | 12 | |||||||||||||
| 2 | 56 | 72 | 16 | 144 | ||||||||||||
| 3 | 360 | 888 | 488 | 64 | 1800 | |||||||||||
| 4 | 2184 | 9032 | 9336 | 2952 | 256 | 23760 | ||||||||||
| 5 | 12712 | 81304 | 139232 | 81560 | 16808 | 1024 | 332640 | |||||||||
| 6 | 71752 | 672360 | 1764288 | 1696720 | 640632 | 92232 | 4096 | 4942080 | ||||||||
| 7 | 395624 | 5225272 | 19913016 | 29150000 | 17945080 | 4698456 | 493928 | 16384 | 77837760 | |||||||
| 8 | 2141960 | 38737416 | 206181928 | 435737144 | 406365720 | 172600120 | 32865464 | 2600712 | 65536 | 1297296000 | ||||||
| 9 | 11430696 | 276815832 | 1997698544 | 5860802152 | 7858765776 | 5039544104 | 1551390448 | 222172056 | 13527848 | 262144 | 22832409600 | |||||
| 10 | 60299208 | 1921179560 | 18368268912 | 72590851008 | 134576090688 | 123743921616 | 57381647760 | 13260150528 | 1464220152 | 69736392 | 1048576 | 423437414400 | ||||
| 11 | 315127528 | 13023023224 | 161931896840 | 841833078720 | 2093157197520 | 2653929015024 | 1761588023952 | 612362553600 | 109063022280 | 9465377240 | 357070568 | 4194304 | 8257029580800 | |||
| 12 | 1634357896 | 86596250952 | 1379406053016 | 9254648226920 | 30125373414480 | 51080550188688 | 46675023846576 | 23202642858768 | 6213002483880 | 870532257320 | 60282722616 | 1818907272 | 16777216 | 168951528345600 | ||
| 13 | 8423447720 | 566755780888 | 11422168821888 | 97359620711224 | 406854553991960 | 900233103944880 | 1098059277373824 | 750766576987824 | 287390565689112 | 60554554755640 | 6785519964800 | 379475787288 | 9228754088 | 67108864 | 3620389893120000 | |
| 14 | 43190980424 | 3660722040552 | 92373670899616 | 987352728354352 | 5212016271239976 | 14751572116831352 | 23430478937137536 | 21362365794309408 | 11243743123836696 | 3387456208942168 | 571361945382624 | 51895054024112 | 2366892832376 | 46680641352 | 268435456 | 81096733605888000 |
Compute more
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Technical details
- Node id
n-b972da6d709613ba· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-23daeb7237195c22
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: 8, 4; 2: 56, 72, 16; 3: 360, 888, 488, 64; 4: 2184, 9032, 9336, 2952, 256; 5: 12712, 81304, 139232, 81560, 16808, 1024; 6: 71752, 672360, 1764288, 1696720, 640632, 92232, 4096; 7: 395624, 5225272, 19913016, 29150000, 17945080, 4698456, 493928, 16384; 8: 2141960, 38737416, 206181928, 435737144, 406365720, 172600120, 32865464, 2600712, 65536; 9: 11430696, 276815832, 1997698544, 5860802152, 7858765776, 5039544104, 1551390448, 222172056, 13527848, 262144;
%F FormulaThe row polynomials satisfy (n+1+2*x)*P(n,x) = -(20*n+40+(8*n^2+20*n+48)*x+(n^3+4*n^2+24*n+16)*x^2+(3*n^2+8*n+12)*x^3+(2*n+4)*x^4)*P(n-2,x) - ((4*n+8)*x+(n^2-n+2)*x^2-(n^2+6)*x^3-(3*n+2)*x^4-2*x^5)*d/dx P(n-2,x) + (9*n+13+(2*n^2+5*n+21)*x+(5*n+6)*x^2+2*x^3)*P(n-1,x) + ((n+1)*x-(n-1)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 8 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 40 + (8*n^2 + 20*n + 48)*x + (n^3 + 4*n^2 + 24*n + 16)*x^2 + (3*n^2 + 8*n + 12)*x^3 + (2*n + 4)*x^4)*P[n - 2, x] - ((4*n + 8)*x + (n^2 - n + 2)*x^2 - (n^2 + 6)*x^3 - (3*n + 2)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (9*n + 13 + (2*n^2 + 5*n + 21)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n + 40 + (8*n^2 + 20*n + 48)*x + (n^3 + 4*n^2 + 24*n + 16)*x^2 + (3*n^2 + 8*n + 12)*x^3 + (2*n + 4)*x^4)*v[i - 2] - ((4*n + 8)*x + (n^2 - n + 2)*x^2 - (n^2 + 6)*x^3 - (3*n + 2)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (9*n + 13 + (2*n^2 + 5*n + 21)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 2*x)); 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 differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
References
- [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.