A046716 → basis A013609 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-8 + 4\,n\right)\,P_{n}(x) = \left(132 + 864\,x + 1872\,x^{2} + 1344\,x^{3} - 116\,n - 792\,n x - 1776\,n x^{2} - 1312\,n x^{3} + 36\,n^{2} + 240\,n^{2} x + 528\,n^{2} x^{2} + 384\,n^{2} x^{3} - 4\,n^{3} - 24\,n^{3} x - 48\,n^{3} x^{2} - 32\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(24 + 204\,x + 648\,x^{2} + 912\,x^{3} + 480\,x^{4} - 23\,n - 200\,n x - 648\,n x^{2} - 928\,n x^{3} - 496\,n x^{4} + 8\,n^{2} + 68\,n^{2} x + 216\,n^{2} x^{2} + 304\,n^{2} x^{3} + 160\,n^{2} x^{4} - n^{3} - 8\,n^{3} x - 24\,n^{3} x^{2} - 32\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-60 - 224\,x - 192\,x^{2} + 52\,n + 208\,n x + 192\,n x^{2} - 12\,n^{2} - 48\,n^{2} x - 48\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8 - 68\,x - 184\,x^{2} - 160\,x^{3} + 13\,n + 96\,n x + 236\,n x^{2} + 192\,n x^{3} - 6\,n^{2} - 42\,n^{2} x - 96\,n^{2} x^{2} - 72\,n^{2} x^{3} + n^{3} + 6\,n^{3} x + 12\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(20 + 64\,x - 24\,n - 64\,n x + 8\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 36\,x - 40\,x^{2} + 5\,n + 24\,n x + 28\,n x^{2} - n^{2} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)
\(P_{3}(x) = 5 + 20\,x + 12\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 2 | 4 | 6 | ||||||
| 3 | 5 | 20 | 12 | 37 | |||||
| 4 | 16 | 100 | 132 | 32 | 280 | ||||
| 5 | 65 | 576 | 1284 | 896 | 80 | 2901 | |||
| 6 | 326 | 3844 | 12648 | 16352 | 7520 | 192 | 40882 | ||
| 7 | 1957 | 29252 | 130128 | 255072 | 232320 | 80832 | 448 | 730009 | |
| 8 | 13700 | 250100 | 1417572 | 3782272 | 5280720 | 3743232 | 1065792 | 1024 | 15554412 |
Compute more
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Technical details
- Node id
n-f8c8c70b62f84626· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-0b269bf170a88844
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, 4; 3: 5, 20, 12; 4: 16, 100, 132, 32; 5: 65, 576, 1284, 896, 80; 6: 326, 3844, 12648, 16352, 7520, 192; 7: 1957, 29252, 130128, 255072, 232320, 80832, 448; 8: 13700, 250100, 1417572, 3782272, 5280720, 3743232, 1065792, 1024; 9: 109601, 2374944, 16456680, 56000256, 106545520, 115994112, 67715200, 16465920, 2304; 10: 986410, 24817476, 204029712, 848215232, 2056320480, 3047049792, 2728178432, 1358960640, 289571328, 5120;
%F FormulaThe row polynomials satisfy (4*n-8)*P(n,x) = -(4*n^3-36*n^2+116*n-132+(24*n^3-240*n^2+792*n-864)*x+(48*n^3-528*n^2+1776*n-1872)*x^2+(32*n^3-384*n^2+1312*n-1344)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(8*n^3-68*n^2+200*n-204)*x+(24*n^3-216*n^2+648*n-648)*x^2+(32*n^3-304*n^2+928*n-912)*x^3+(16*n^3-160*n^2+496*n-480)*x^4)*d/dx P(n-3,x) - (12*n^2-52*n+60+(48*n^2-208*n+224)*x+(48*n^2-192*n+192)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(6*n^3-42*n^2+96*n-68)*x+(12*n^3-96*n^2+236*n-184)*x^2+(8*n^3-72*n^2+192*n-160)*x^3)*d/dx P(n-2,x) + (8*n^2-24*n+20+(16*n^2-64*n+64)*x)*P(n-1,x) - (n^2-5*n+8+(4*n^2-24*n+36)*x+(4*n^2-28*n+40)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 4*x, P(3,x) = 5 + 20*x + 12*x^2.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[3, x_] := 5 + 20*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 36*n^2 + 116*n - 132 + (24*n^3 - 240*n^2 + 792*n - 864)*x + (48*n^3 - 528*n^2 + 1776*n - 1872)*x^2 + (32*n^3 - 384*n^2 + 1312*n - 1344)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*D[P[n - 3, x], x] - (12*n^2 - 52*n + 60 + (48*n^2 - 208*n + 224)*x + (48*n^2 - 192*n + 192)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 24*n + 20 + (16*n^2 - 64*n + 64)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 5 + 20*x + 12*x^2; for(n = 4, N, my(i = n); v[i] = (-(4*n^3 - 36*n^2 + 116*n - 132 + (24*n^3 - 240*n^2 + 792*n - 864)*x + (48*n^3 - 528*n^2 + 1776*n - 1872)*x^2 + (32*n^3 - 384*n^2 + 1312*n - 1344)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*deriv(v[i - 3], x) - (12*n^2 - 52*n + 60 + (48*n^2 - 208*n + 224)*x + (48*n^2 - 192*n + 192)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 24*n + 20 + (16*n^2 - 64*n + 64)*x)*v[i - 1] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*deriv(v[i - 1], x))/(4*n - 8)); 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_A013609 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_A013609 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