A039756 → x ↦ x+1 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(-6 - 26\,x - 21\,x^{2} + 22\,x^{3} + 24\,x^{4} + 20\,n + 68\,n x + 78\,n x^{2} + 16\,n x^{3} - 16\,n x^{4} + 4\,n^{2} - 8\,n^{2} x - 36\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} - 16\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 32\,x^{2} + 62\,x^{3} + 52\,x^{4} + 16\,x^{5} + 12\,n + 56\,n x + 100\,n x^{2} + 80\,n x^{3} + 24\,n x^{4} + 8\,n^{2} + 24\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 - 18\,x - 10\,x^{2} + 8\,x^{3} + 16\,n x + 20\,n x^{2} + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 - 12\,x - 18\,x^{2} - 8\,x^{3} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 1 | 4 | ||||||
| 2 | 12 | 8 | 1 | 21 | |||||
| 3 | 62 | 70 | 19 | 1 | 152 | ||||
| 4 | 372 | 624 | 316 | 48 | 1 | 1361 | |||
| 5 | 2492 | 5636 | 4368 | 1328 | 131 | 1 | 13956 | ||
| 6 | 18384 | 52768 | 56028 | 26688 | 5456 | 376 | 1 | 159701 | |
| 7 | 147800 | 517336 | 705500 | 468548 | 153926 | 22398 | 1107 | 1 | 2016616 |
Compute more
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Technical details
- Node id
n-9e459b82002bdc2e· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-98d00330e16c9680
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: 3, 1; 2: 12, 8, 1; 3: 62, 70, 19, 1; 4: 372, 624, 316, 48, 1; 5: 2492, 5636, 4368, 1328, 131, 1; 6: 18384, 52768, 56028, 26688, 5456, 376, 1; 7: 147800, 517336, 705500, 468548, 153926, 22398, 1107, 1; 8: 1283088, 5321088, 8926048, 7710720, 3602056, 869216, 93192, 3296, 1; 9: 11939376, 57393040, 114784768, 123034880, 75727928, 26533064, 4895920, 396784, 9859, 1;
%F FormulaThe row polynomials satisfy (2*n+1+4*x)*P(n,x) = -(8*n^3-4*n^2-20*n+6+(16*n^3+8*n^2-68*n+26)*x+(8*n^3+36*n^2-78*n+21)*x^2+(24*n^2-16*n-22)*x^3+(16*n-24)*x^4)*P(n-2,x) + (8*n^2+12*n+(24*n^2+56*n+6)*x+(24*n^2+100*n+32)*x^2+(8*n^2+80*n+62)*x^3+(24*n+52)*x^4+16*x^5)*d/dx P(n-2,x) + (8*n^2-4+(8*n^2+16*n-18)*x+(20*n-10)*x^2+8*x^3)*P(n-1,x) - (4*n+2+(8*n+12)*x+(4*n+18)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 4*n^2 - 20*n + 6 + (16*n^3 + 8*n^2 - 68*n + 26)*x + (8*n^3 + 36*n^2 - 78*n + 21)*x^2 + (24*n^2 - 16*n - 22)*x^3 + (16*n - 24)*x^4)*P[n - 2, x] + (8*n^2 + 12*n + (24*n^2 + 56*n + 6)*x + (24*n^2 + 100*n + 32)*x^2 + (8*n^2 + 80*n + 62)*x^3 + (24*n + 52)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (8*n^2 - 4 + (8*n^2 + 16*n - 18)*x + (20*n - 10)*x^2 + 8*x^3)*P[n - 1, x] - (4*n + 2 + (8*n + 12)*x + (4*n + 18)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(2*n + 1 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 4*n^2 - 20*n + 6 + (16*n^3 + 8*n^2 - 68*n + 26)*x + (8*n^3 + 36*n^2 - 78*n + 21)*x^2 + (24*n^2 - 16*n - 22)*x^3 + (16*n - 24)*x^4)*v[i - 2] + (8*n^2 + 12*n + (24*n^2 + 56*n + 6)*x + (24*n^2 + 100*n + 32)*x^2 + (8*n^2 + 80*n + 62)*x^3 + (24*n + 52)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (8*n^2 - 4 + (8*n^2 + 16*n - 18)*x + (20*n - 10)*x^2 + 8*x^3)*v[i - 1] - (4*n + 2 + (8*n + 12)*x + (4*n + 18)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(2*n + 1 + 4*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 differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
shift_x_plus_one of Triangle of B-analogs of Stirling numbers of 2nd kind.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of Triangle of B-analogs of Stirling numbers of 2nd kind.?
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.