A046716 → basis A013610 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-12 + 6\,n\right)\,P_{n}(x) = \left(198 + 1944\,x + 6318\,x^{2} + 6804\,x^{3} - 174\,n - 1782\,n x - 5994\,n x^{2} - 6642\,n x^{3} + 54\,n^{2} + 540\,n^{2} x + 1782\,n^{2} x^{2} + 1944\,n^{2} x^{3} - 6\,n^{3} - 54\,n^{3} x - 162\,n^{3} x^{2} - 162\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(24 + 306\,x + 1458\,x^{2} + 3078\,x^{3} + 2430\,x^{4} - 23\,n - 300\,n x - 1458\,n x^{2} - 3132\,n x^{3} - 2511\,n x^{4} + 8\,n^{2} + 102\,n^{2} x + 486\,n^{2} x^{2} + 1026\,n^{2} x^{3} + 810\,n^{2} x^{4} - n^{3} - 12\,n^{3} x - 54\,n^{3} x^{2} - 108\,n^{3} x^{3} - 81\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-90 - 504\,x - 648\,x^{2} + 78\,n + 468\,n x + 648\,n x^{2} - 18\,n^{2} - 108\,n^{2} x - 162\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8 - 102\,x - 414\,x^{2} - 540\,x^{3} + 13\,n + 144\,n x + 531\,n x^{2} + 648\,n x^{3} - 6\,n^{2} - 63\,n^{2} x - 216\,n^{2} x^{2} - 243\,n^{2} x^{3} + n^{3} + 9\,n^{3} x + 27\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(30 + 144\,x - 36\,n - 144\,n x + 12\,n^{2} + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 54\,x - 90\,x^{2} + 5\,n + 36\,n x + 63\,n x^{2} - n^{2} - 6\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 6\,x\)
\(P_{3}(x) = 5 + 30\,x + 27\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 2 | 6 | 8 | ||||||
| 3 | 5 | 30 | 27 | 62 | |||||
| 4 | 16 | 150 | 297 | 108 | 571 | ||||
| 5 | 65 | 864 | 2889 | 3024 | 405 | 7247 | |||
| 6 | 326 | 5766 | 28458 | 55188 | 38070 | 1458 | 129266 | ||
| 7 | 1957 | 43878 | 292788 | 860868 | 1176120 | 613818 | 5103 | 2994532 | |
| 8 | 13700 | 375150 | 3189537 | 12765168 | 26733645 | 28425168 | 12140037 | 17496 | 83659901 |
Compute more
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Technical details
- Node id
n-a9aff647c13fb176· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-90434d8020f51a22
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: 2, 6; 3: 5, 30, 27; 4: 16, 150, 297, 108; 5: 65, 864, 2889, 3024, 405; 6: 326, 5766, 28458, 55188, 38070, 1458; 7: 1957, 43878, 292788, 860868, 1176120, 613818, 5103; 8: 13700, 375150, 3189537, 12765168, 26733645, 28425168, 12140037, 17496; 9: 109601, 3562416, 37027530, 189000864, 539386695, 880830288, 771318450, 281335680, 59049; 10: 986410, 37226214, 459066852, 2862726408, 10410122430, 23138534358, 31075657452, 23219116560, 7421396418, 196830;
%F FormulaThe row polynomials satisfy (6*n-12)*P(n,x) = -(6*n^3-54*n^2+174*n-198+(54*n^3-540*n^2+1782*n-1944)*x+(162*n^3-1782*n^2+5994*n-6318)*x^2+(162*n^3-1944*n^2+6642*n-6804)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(12*n^3-102*n^2+300*n-306)*x+(54*n^3-486*n^2+1458*n-1458)*x^2+(108*n^3-1026*n^2+3132*n-3078)*x^3+(81*n^3-810*n^2+2511*n-2430)*x^4)*d/dx P(n-3,x) - (18*n^2-78*n+90+(108*n^2-468*n+504)*x+(162*n^2-648*n+648)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(9*n^3-63*n^2+144*n-102)*x+(27*n^3-216*n^2+531*n-414)*x^2+(27*n^3-243*n^2+648*n-540)*x^3)*d/dx P(n-2,x) + (12*n^2-36*n+30+(36*n^2-144*n+144)*x)*P(n-1,x) - (n^2-5*n+8+(6*n^2-36*n+54)*x+(9*n^2-63*n+90)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 6*x, P(3,x) = 5 + 30*x + 27*x^2.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[3, x_] := 5 + 30*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^3 - 54*n^2 + 174*n - 198 + (54*n^3 - 540*n^2 + 1782*n - 1944)*x + (162*n^3 - 1782*n^2 + 5994*n - 6318)*x^2 + (162*n^3 - 1944*n^2 + 6642*n - 6804)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*D[P[n - 3, x], x] - (18*n^2 - 78*n + 90 + (108*n^2 - 468*n + 504)*x + (162*n^2 - 648*n + 648)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*D[P[n - 2, x], x] + (12*n^2 - 36*n + 30 + (36*n^2 - 144*n + 144)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*D[P[n - 1, x], x])/(6*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; v[3] = 5 + 30*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-(6*n^3 - 54*n^2 + 174*n - 198 + (54*n^3 - 540*n^2 + 1782*n - 1944)*x + (162*n^3 - 1782*n^2 + 5994*n - 6318)*x^2 + (162*n^3 - 1944*n^2 + 6642*n - 6804)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*deriv(v[i - 3], x) - (18*n^2 - 78*n + 90 + (108*n^2 - 468*n + 504)*x + (162*n^2 - 648*n + 648)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*deriv(v[i - 2], x) + (12*n^2 - 36*n + 30 + (36*n^2 - 144*n + 144)*x)*v[i - 1] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*deriv(v[i - 1], x))/(6*n - 12)); 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]
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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
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, 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
basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.?
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Geometric basis map (elementary) polytool ? step ›
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from
Coefficients of a special case of Poisson-Charlier polynomials.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Coefficients of a special case of Poisson-Charlier polynomials.?
- ↑ Three-term recurrence criterion (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Coefficients of a special case of Poisson-Charlier polynomials.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Coefficients of a special case of Poisson-Charlier polynomials.?
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from
basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023