A090582 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n^{2}\right)\,P_{n}(x) = \left(4\,x - n^{2} - 7\,n^{2} x^{2} - 2\,n^{3} - 2\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(x + 5\,x^{2} + n^{2} x - 4\,n^{2} x^{2} - 13\,n^{2} x^{3} + 2\,n^{3} x + 4\,n^{3} x^{2} - 14\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{2} + x^{3} - n^{2} x^{2} - 4\,n^{2} x^{3} - 3\,n^{2} x^{4} - 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(n + 3\,n^{2} + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 2\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 4 | 6 | ||||||
| 2 | 6 | 24 | 9 | 39 | |||||
| 3 | 24 | 144 | 126 | 16 | 310 | ||||
| 4 | 120 | 960 | 1350 | 480 | 25 | 2935 | |||
| 5 | 720 | 7200 | 14040 | 8640 | 1550 | 36 | 32186 | ||
| 6 | 5040 | 60480 | 151200 | 134400 | 45150 | 4536 | 49 | 400855 | |
| 7 | 40320 | 564480 | 1723680 | 2016000 | 1020600 | 208656 | 12446 | 64 | 5586246 |
Compute more
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Technical details
- Node id
n-5ddd616bf530de21· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-856ad640314d7fe1
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: 2, 4; 2: 6, 24, 9; 3: 24, 144, 126, 16; 4: 120, 960, 1350, 480, 25; 5: 720, 7200, 14040, 8640, 1550, 36; 6: 5040, 60480, 151200, 134400, 45150, 4536, 49; 7: 40320, 564480, 1723680, 2016000, 1020600, 208656, 12446, 64; 8: 362880, 5806080, 20956320, 30481920, 20853000, 6713280, 889350, 32640, 81; 9: 3628800, 65318400, 272160000, 474163200, 410886000, 183708000, 40107480, 3582720, 82782, 100;
%F FormulaThe row polynomials satisfy n^2*P(n,x) = -(n^4+2*n^3+n^2+(4*n^4-4)*x-(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) + ((2*n^3+n^2+1)*x+(4*n^4+4*n^3-4*n^2+5)*x^2+(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) - ((n^2-1)*x^2+(4*n^3+4*n^2-1)*x^3+(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3+3*n^2+n+4*n^3*x)*P(n-1,x) - ((2*n^2-1)*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) = 2 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 + n^2 + (4*n^4 - 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] + ((2*n^3 + n^2 + 1)*x + (4*n^4 + 4*n^3 - 4*n^2 + 5)*x^2 + (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 - 1)*x^2 + (4*n^3 + 4*n^2 - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 + 3*n^2 + n + 4*n^3*x)*P[n - 1, x] - ((2*n^2 - 1)*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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 + n^2 + (4*n^4 - 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] + ((2*n^3 + n^2 + 1)*x + (4*n^4 + 4*n^3 - 4*n^2 + 5)*x^2 + (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 - 1)*x^2 + (4*n^3 + 4*n^2 - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 + 3*n^2 + n + 4*n^3*x)*v[i - 1] - ((2*n^2 - 1)*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
T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= …?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= …?
Related ?
Same fingerprint ?
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
- [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.