A062190 → reverse → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(80\,n + 16\,n^{2} - 40\,n^{2} x + 47\,n^{3} x + 11\,n^{4} x + 5\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(400 - 320\,x + 40\,x^{2} + 160\,n + 312\,n x - 164\,n x^{2} + 16\,n^{2} + 387\,n^{2} x - 26\,n^{2} x^{2} - 15\,n^{2} x^{3} + 102\,n^{3} x + 168\,n^{3} x^{2} - 4\,n^{3} x^{3} + 11\,n^{4} x + 79\,n^{4} x^{2} + 24\,n^{4} x^{3} + 10\,n^{5} x^{2} + 16\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(80\,x - 100\,x^{2} + 20\,x^{3} + 32\,n x + 30\,n x^{2} - 62\,n x^{3} + 72\,n^{2} x^{2} - 57\,n^{2} x^{3} - 15\,n^{2} x^{4} + 22\,n^{3} x^{2} + 2\,n^{3} x^{3} - 24\,n^{3} x^{4} + 11\,n^{4} x^{3} - 11\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 6 | 4 | 10 | ||||||
| 2 | 21 | 56 | 9 | 86 | |||||
| 3 | 56 | 336 | 216 | 16 | 624 | ||||
| 4 | 126 | 1344 | 1944 | 576 | 25 | 4015 | |||
| 5 | 252 | 4200 | 10800 | 7200 | 1250 | 36 | 23738 | ||
| 6 | 462 | 11088 | 44550 | 52800 | 20625 | 2376 | 49 | 131950 | |
| 7 | 792 | 25872 | 149688 | 277200 | 192500 | 49896 | 4116 | 64 | 700128 |
Compute more
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Technical details
- Node id
n-7c9ee23115755ba4· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-8f9b1f1bf179778d
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: 6, 4; 2: 21, 56, 9; 3: 56, 336, 216, 16; 4: 126, 1344, 1944, 576, 25; 5: 252, 4200, 10800, 7200, 1250, 36; 6: 462, 11088, 44550, 52800, 20625, 2376, 49; 7: 792, 25872, 149688, 277200, 192500, 49896, 4116, 64; 8: 1287, 54912, 432432, 1153152, 1251250, 576576, 107016, 6656, 81; 9: 2002, 108108, 1111968, 4036032, 6306300, 4540536, 1498224, 209664, 10206, 100;
%F FormulaThe row polynomials satisfy (16*n^2+80*n+(11*n^4+47*n^3-40*n^2)*x+(n^6+5*n^5)*x^2)*P(n,x) = (16*n^2+160*n+400+(11*n^4+102*n^3+387*n^2+312*n-320)*x+(n^6+10*n^5+79*n^4+168*n^3-26*n^2-164*n+40)*x^2+(3*n^6+16*n^5+24*n^4-4*n^3-15*n^2)*x^3)*P(n-1,x) + ((32*n+80)*x+(22*n^3+72*n^2+30*n-100)*x^2+(2*n^5+11*n^4+2*n^3-57*n^2-62*n+20)*x^3-(2*n^5+11*n^4+24*n^3+15*n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 + 160*n + 400 + (11*n^4 + 102*n^3 + 387*n^2 + 312*n - 320)*x + (n^6 + 10*n^5 + 79*n^4 + 168*n^3 - 26*n^2 - 164*n + 40)*x^2 + (3*n^6 + 16*n^5 + 24*n^4 - 4*n^3 - 15*n^2)*x^3)*P[n - 1, x] + ((32*n + 80)*x + (22*n^3 + 72*n^2 + 30*n - 100)*x^2 + (2*n^5 + 11*n^4 + 2*n^3 - 57*n^2 - 62*n + 20)*x^3 - (2*n^5 + 11*n^4 + 24*n^3 + 15*n^2)*x^4)*D[P[n - 1, x], x])/(16*n^2 + 80*n + (11*n^4 + 47*n^3 - 40*n^2)*x + (n^6 + 5*n^5)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^2 + 160*n + 400 + (11*n^4 + 102*n^3 + 387*n^2 + 312*n - 320)*x + (n^6 + 10*n^5 + 79*n^4 + 168*n^3 - 26*n^2 - 164*n + 40)*x^2 + (3*n^6 + 16*n^5 + 24*n^4 - 4*n^3 - 15*n^2)*x^3)*v[i - 1] + ((32*n + 80)*x + (22*n^3 + 72*n^2 + 30*n - 100)*x^2 + (2*n^5 + 11*n^4 + 2*n^3 - 57*n^2 - 62*n + 20)*x^3 - (2*n^5 + 11*n^4 + 24*n^3 + 15*n^2)*x^4)*deriv(v[i - 1], x))/(16*n^2 + 80*n + (11*n^4 + 47*n^3 - 40*n^2)*x + (n^6 + 5*n^5)*x^2)); 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
reversal of Coefficient triangle of certain polynomials N(5; m,x).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Coefficient triangle of certain polynomials N(5; m,x).?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Coefficient triangle of certain polynomials N(5; m,x).?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Coefficient triangle of certain polynomials N(5; m,x).?
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.