A108426 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2\,n + 2\,n^{2} + 12\,n^{3} + 4\,n^{3} x + 8\,n^{4} x\right)\,P_{n}(x) = \left(-6 - 10\,x - 4\,x^{2} + 45\,n + 76\,n x + 31\,n x^{2} + 2\,n x^{3} - 108\,n^{2} - 174\,n^{2} x - 66\,n^{2} x^{2} - 2\,n^{2} x^{3} + 81\,n^{3} + 90\,n^{3} x + 15\,n^{3} x^{2} - 2\,n^{3} x^{3} + 54\,n^{4} x + 48\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-3\,x - 5\,x^{2} - 2\,x^{3} + 18\,n x + 35\,n x^{2} + 19\,n x^{3} + 2\,n x^{4} - 27\,n^{2} x - 48\,n^{2} x^{2} - 21\,n^{2} x^{3} - 18\,n^{3} x^{2} - 20\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 3 | 10 | 6 | 19 | |||||
| 3 | 12 | 56 | 63 | 20 | 151 | ||||
| 4 | 55 | 330 | 540 | 336 | 70 | 1331 | |||
| 5 | 273 | 2002 | 4290 | 3960 | 1650 | 252 | 12427 | ||
| 6 | 1428 | 12376 | 32760 | 40040 | 25025 | 7722 | 924 | 120275 | |
| 7 | 7752 | 77520 | 244188 | 371280 | 309400 | 144144 | 35035 | 3432 | 1192751 |
Compute more
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Technical details
- Node id
n-1e24992b6bf1f3cd· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-51b62033f048530c
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: 1, 2; 2: 3, 10, 6; 3: 12, 56, 63, 20; 4: 55, 330, 540, 336, 70; 5: 273, 2002, 4290, 3960, 1650, 252; 6: 1428, 12376, 32760, 40040, 25025, 7722, 924; 7: 7752, 77520, 244188, 371280, 309400, 144144, 35035, 3432; 8: 43263, 490314, 1790712, 3255840, 3391500, 2116296, 779688, 155584, 12870; 9: 246675, 3124550, 12978900, 27457584, 34321980, 26860680, 13294680, 4031040, 680238, 48620;
%F FormulaThe row polynomials satisfy (12*n^3+2*n^2-2*n+(8*n^4+4*n^3)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(54*n^4+90*n^3-174*n^2+76*n-10)*x+(48*n^4+15*n^3-66*n^2+31*n-4)*x^2+(2*n^4-2*n^3-2*n^2+2*n)*x^3)*P(n-1,x) - ((27*n^2-18*n+3)*x+(18*n^3+48*n^2-35*n+5)*x^2+(20*n^3+21*n^2-19*n+2)*x^3+(2*n^3-2*n)*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[((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 90*n^3 - 174*n^2 + 76*n - 10)*x + (48*n^4 + 15*n^3 - 66*n^2 + 31*n - 4)*x^2 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^3)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 48*n^2 - 35*n + 5)*x^2 + (20*n^3 + 21*n^2 - 19*n + 2)*x^3 + (2*n^3 - 2*n)*x^4)*D[P[n - 1, x], x])/(12*n^3 + 2*n^2 - 2*n + (8*n^4 + 4*n^3)*x)]];
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] = ((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 90*n^3 - 174*n^2 + 76*n - 10)*x + (48*n^4 + 15*n^3 - 66*n^2 + 31*n - 4)*x^2 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^3)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 48*n^2 - 35*n + 5)*x^2 + (20*n^3 + 21*n^2 - 19*n + 2)*x^3 + (2*n^3 - 2*n)*x^4)*deriv(v[i - 1], x))/(12*n^3 + 2*n^2 - 2*n + (8*n^4 + 4*n^3)*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
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
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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
Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
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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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from
Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
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from
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.