theta_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - 2\,x + 2\,n x\right)\,P_{n}(x) = \left(-6\,x^{2} + 12\,x^{3} - 6\,x^{4} + 2\,n x^{2} - 20\,n x^{3} + 34\,n x^{4} - 12\,n x^{5} + 4\,n^{2} x^{3} - 20\,n^{2} x^{4} + 20\,n^{2} x^{5} - 8\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 2\,x^{3} + 2\,x^{4} - 2\,x^{5} - 6\,n x^{3} + 4\,n x^{4} + 6\,n x^{5} - 4\,n x^{6} - 4\,n^{2} x^{4} + 4\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(1 + 2\,x - 9\,x^{2} + 6\,x^{3} + 2\,n x + 12\,n x^{2} - 14\,n x^{3} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 2\,x^{2} - x^{3} + 2\,x^{4} + 2\,n x^{2} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x + 3\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 4 | 3 | 8 | ||||||||||||
| 2 | 1 | 12 | 30 | 24 | 5 | 72 | ||||||||||
| 3 | 1 | 28 | 141 | 272 | 235 | 84 | 7 | 768 | ||||||||
| 4 | 1 | 60 | 528 | 1800 | 3030 | 2700 | 1232 | 240 | 9 | 9600 | ||||||
| 5 | 1 | 124 | 1791 | 9568 | 25810 | 39672 | 36134 | 19136 | 5373 | 620 | 11 | 138240 | ||||
| 6 | 1 | 252 | 5778 | 45528 | 179835 | 414792 | 598444 | 553056 | 323703 | 113820 | 21186 | 1512 | 13 | 2257920 | ||
| 7 | 1 | 508 | 18129 | 203568 | 1118925 | 3603276 | 7422877 | 10216768 | 9543699 | 6005460 | 2461635 | 610704 | 78559 | 3556 | 15 | 41287680 |
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Technical details
- Node id
n-7b7663b683c07e49· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-880bd7c6b138382f
Observed on the stored rows ?
- Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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, 3; 2: 1, 12, 30, 24, 5; 3: 1, 28, 141, 272, 235, 84, 7; 4: 1, 60, 528, 1800, 3030, 2700, 1232, 240, 9; 5: 1, 124, 1791, 9568, 25810, 39672, 36134, 19136, 5373, 620, 11; 6: 1, 252, 5778, 45528, 179835, 414792, 598444, 553056, 323703, 113820, 21186, 1512, 13; 7: 1, 508, 18129, 203568, 1118925, 3603276, 7422877, 10216768, 9543699, 6005460, 2461635, 610704, 78559, 3556, 15; 8: 1, 1020, 55956, 876840, 6496340, 27969300, 77599228, 146744400, 194774310, 183430500, 121941644, 55938600, 16890484, 3068940, 279780, 8160, 17; 9: 1, 2044, 170979, 3689216, 36072700, 201820656, 726689684, 1798833664, 3180618486, 4101441800, 3887422594, 2698250496, 1349566556, 470914864, 108218100, 14756864, 968881, 18396, 19;
%F FormulaThe row polynomials satisfy (1+(2*n-2)*x)*P(n,x) = ((2*n-6)*x^2+(4*n^2-20*n+12)*x^3-(20*n^2-34*n+6)*x^4-(8*n^3-20*n^2+12*n)*x^5)*P(n-2,x) - (2*x^2+(6*n-2)*x^3+(4*n^2-4*n-2)*x^4-(6*n-2)*x^5-(4*n^2-4*n)*x^6)*d/dx P(n-2,x) + (1+(2*n+2)*x+(12*n-9)*x^2+(8*n^2-14*n+6)*x^3)*P(n-1,x) + (x+(2*n-2)*x^2-x^3-(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x + 3*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 6)*x^2 + (4*n^2 - 20*n + 12)*x^3 - (20*n^2 - 34*n + 6)*x^4 - (8*n^3 - 20*n^2 + 12*n)*x^5)*P[n - 2, x] - (2*x^2 + (6*n - 2)*x^3 + (4*n^2 - 4*n - 2)*x^4 - (6*n - 2)*x^5 - (4*n^2 - 4*n)*x^6)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x + (12*n - 9)*x^2 + (8*n^2 - 14*n + 6)*x^3)*P[n - 1, x] + (x + (2*n - 2)*x^2 - x^3 - (2*n - 2)*x^4)*D[P[n - 1, x], x])/(1 + (2*n - 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] = 1 + 4*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 6)*x^2 + (4*n^2 - 20*n + 12)*x^3 - (20*n^2 - 34*n + 6)*x^4 - (8*n^3 - 20*n^2 + 12*n)*x^5)*v[i - 2] - (2*x^2 + (6*n - 2)*x^3 + (4*n^2 - 4*n - 2)*x^4 - (6*n - 2)*x^5 - (4*n^2 - 4*n)*x^6)*deriv(v[i - 2], x) + (1 + (2*n + 2)*x + (12*n - 9)*x^2 + (8*n^2 - 14*n + 6)*x^3)*v[i - 1] + (x + (2*n - 2)*x^2 - x^3 - (2*n - 2)*x^4)*deriv(v[i - 1], x))/(1 + (2*n - 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]
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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
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
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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Product of real-rooted factors (elementary) polytool ? step ›
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
- ↑ Product of real-rooted factors (elementary) (above)
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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.