type_b_eulerian → γ (scale 1/4) → x ↦ x+1 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n^{2}\right)\,P_{n}(x) = \left(16\,x + n + n^{2} - 4\,n^{2} x - 10\,n^{2} x^{2} - n^{3} + 8\,n^{3} x - 13\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 20\,x^{2} - 2\,n^{2} x - 20\,n^{2} x^{2} - 18\,n^{2} x^{3} + 4\,n^{3} x - 32\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{2} + 4\,x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3} - 4\,n^{2} x^{4} - 8\,n^{3} x^{3} - 8\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(n + 2\,n^{2} - 4\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 4\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 1 | |||||||
| 2 | 2 | 4 | 6 | ||||||
| 3 | 6 | 20 | 26 | ||||||
| 4 | 24 | 112 | 45 | 181 | |||||
| 5 | 120 | 720 | 549 | 1389 | |||||
| 6 | 720 | 5280 | 5958 | 976 | 12934 | ||||
| 7 | 5040 | 43680 | 65394 | 22160 | 136274 | ||||
| 8 | 40320 | 403200 | 752976 | 393088 | 34625 | 1624209 | |||
| 9 | 362880 | 4112640 | 9208080 | 6533760 | 1263025 | 21480385 | |||
| 10 | 3628800 | 45964800 | 120022560 | 107984640 | 33153050 | 1818756 | 312572606 | ||
| 11 | 39916800 | 558835200 | 1667524320 | 1820163840 | 774363150 | 97299540 | 4958102850 | ||
| 12 | 479001600 | 7344691200 | 24656607360 | 31672396800 | 17317482600 | 3539847888 | 132435485 | 85142462933 | |
| 13 | 6227020800 | 103783680000 | 387165018240 | 572378526720 | 383516133000 | 110368243440 | 9768688069 | 1573207310269 | |
| 14 | 87178291200 | 1569209241600 | 6440296262400 | 10773668505600 | 8561486934000 | 3202789131360 | 470207703518 | 12759102784 | 31117595172462 |
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Technical details
- Node id
n-b47eb1d248b2ecb3· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-afd5149745c5a1a0
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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, 4; 3: 6, 20; 4: 24, 112, 45; 5: 120, 720, 549; 6: 720, 5280, 5958, 976; 7: 5040, 43680, 65394, 22160; 8: 40320, 403200, 752976, 393088, 34625; 9: 362880, 4112640, 9208080, 6533760, 1263025;
%F FormulaThe row polynomials satisfy n^2*P(n,x) = -(n^4+n^3-n^2-n+(4*n^4-8*n^3+4*n^2-16)*x-(9*n^4-13*n^3-10*n^2)*x^2)*P(n-2,x) + ((4*n^3-2*n^2+6)*x+(4*n^4-20*n^2+20)*x^2+(4*n^4-32*n^3-18*n^2)*x^3)*d/dx P(n-2,x) - ((4*n^2-4)*x^2+(8*n^3+8*n^2-4)*x^3+(8*n^3+4*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3+2*n^2+n+(4*n^3-4*n^2)*x)*P(n-1,x) - ((4*n^2-2)*x+(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + n^3 - n^2 - n + (4*n^4 - 8*n^3 + 4*n^2 - 16)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*P[n - 2, x] + ((4*n^3 - 2*n^2 + 6)*x + (4*n^4 - 20*n^2 + 20)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4)*x^2 + (8*n^3 + 8*n^2 - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 + 2*n^2 + n + (4*n^3 - 4*n^2)*x)*P[n - 1, x] - ((4*n^2 - 2)*x + (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
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; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + n^3 - n^2 - n + (4*n^4 - 8*n^3 + 4*n^2 - 16)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*v[i - 2] + ((4*n^3 - 2*n^2 + 6)*x + (4*n^4 - 20*n^2 + 20)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4)*x^2 + (8*n^3 + 8*n^2 - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 + 2*n^2 + n + (4*n^3 - 4*n^2)*x)*v[i - 1] - ((4*n^2 - 2)*x + (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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
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
shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length…?
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.