A249250 → reverse → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 - 4\,x - 6\,x^{2} - 4\,x^{3} - x^{4} + n + 4\,n x + 6\,n x^{2} + 4\,n x^{3} + n x^{4}\right)\,P_{n}(x) = \left(-1 + 40\,x^{2} + 164\,x^{3} + 291\,x^{4} + 268\,x^{5} + 126\,x^{6} + 24\,x^{7} + n - 4\,n x - 68\,n x^{2} - 248\,n x^{3} - 427\,n x^{4} - 392\,n x^{5} - 186\,n x^{6} - 36\,n x^{7} + 4\,n^{2} x + 28\,n^{2} x^{2} + 84\,n^{2} x^{3} + 136\,n^{2} x^{4} + 124\,n^{2} x^{5} + 60\,n^{2} x^{6} + 12\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(x + 35\,x^{2} + 208\,x^{3} + 550\,x^{4} + 785\,x^{5} + 631\,x^{6} + 270\,x^{7} + 48\,x^{8} - n x - 18\,n x^{2} - 98\,n x^{3} - 256\,n x^{4} - 369\,n x^{5} - 302\,n x^{6} - 132\,n x^{7} - 24\,n x^{8}\right)\,P_{n-1}'(x) + \left(x^{2} + 17\,x^{3} + 82\,x^{4} + 190\,x^{5} + 245\,x^{6} + 181\,x^{7} + 72\,x^{8} + 12\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 4 | 7 | ||||||
| 2 | 3 | 12 | 10 | 25 | |||||
| 3 | 3 | 24 | 48 | 28 | 103 | ||||
| 4 | 3 | 36 | 126 | 168 | 76 | 409 | |||
| 5 | 3 | 48 | 252 | 564 | 564 | 208 | 1639 | ||
| 6 | 3 | 60 | 426 | 1392 | 2280 | 1824 | 568 | 6553 | |
| 7 | 3 | 72 | 648 | 2844 | 6696 | 8640 | 5760 | 1552 | 26215 |
Compute more
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Technical details
- Node id
n-81398ca66c219916· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-aecab2aaf9adbf77
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, 4; 2: 3, 12, 10; 3: 3, 24, 48, 28; 4: 3, 36, 126, 168, 76; 5: 3, 48, 252, 564, 564, 208; 6: 3, 60, 426, 1392, 2280, 1824, 568; 7: 3, 72, 648, 2844, 6696, 8640, 5760, 1552; 8: 3, 84, 918, 5112, 16020, 29376, 31248, 17856, 4240; 9: 3, 96, 1236, 8388, 33228, 80496, 120672, 109152, 54576, 11584;
%F FormulaThe row polynomials satisfy (n-1+(4*n-4)*x+(6*n-6)*x^2+(4*n-4)*x^3+(n-1)*x^4)*P(n,x) = (n-1+(4*n^2-4*n)*x+(28*n^2-68*n+40)*x^2+(84*n^2-248*n+164)*x^3+(136*n^2-427*n+291)*x^4+(124*n^2-392*n+268)*x^5+(60*n^2-186*n+126)*x^6+(12*n^2-36*n+24)*x^7)*P(n-1,x) - ((n-1)*x+(18*n-35)*x^2+(98*n-208)*x^3+(256*n-550)*x^4+(369*n-785)*x^5+(302*n-631)*x^6+(132*n-270)*x^7+(24*n-48)*x^8)*d/dx P(n-1,x) + (x^2+17*x^3+82*x^4+190*x^5+245*x^6+181*x^7+72*x^8+12*x^9)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (4*n^2 - 4*n)*x + (28*n^2 - 68*n + 40)*x^2 + (84*n^2 - 248*n + 164)*x^3 + (136*n^2 - 427*n + 291)*x^4 + (124*n^2 - 392*n + 268)*x^5 + (60*n^2 - 186*n + 126)*x^6 + (12*n^2 - 36*n + 24)*x^7)*P[n - 1, x] - ((n - 1)*x + (18*n - 35)*x^2 + (98*n - 208)*x^3 + (256*n - 550)*x^4 + (369*n - 785)*x^5 + (302*n - 631)*x^6 + (132*n - 270)*x^7 + (24*n - 48)*x^8)*D[P[n - 1, x], x] + (x^2 + 17*x^3 + 82*x^4 + 190*x^5 + 245*x^6 + 181*x^7 + 72*x^8 + 12*x^9)*D[P[n - 1, x], {x, 2}])/(n - 1 + (4*n - 4)*x + (6*n - 6)*x^2 + (4*n - 4)*x^3 + (n - 1)*x^4)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (4*n^2 - 4*n)*x + (28*n^2 - 68*n + 40)*x^2 + (84*n^2 - 248*n + 164)*x^3 + (136*n^2 - 427*n + 291)*x^4 + (124*n^2 - 392*n + 268)*x^5 + (60*n^2 - 186*n + 126)*x^6 + (12*n^2 - 36*n + 24)*x^7)*v[i - 1] - ((n - 1)*x + (18*n - 35)*x^2 + (98*n - 208)*x^3 + (256*n - 550)*x^4 + (369*n - 785)*x^5 + (302*n - 631)*x^6 + (132*n - 270)*x^7 + (24*n - 48)*x^8)*deriv(v[i - 1], x) + (x^2 + 17*x^3 + 82*x^4 + 190*x^5 + 245*x^6 + 181*x^7 + 72*x^8 + 12*x^9)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (4*n - 4)*x + (6*n - 6)*x^2 + (4*n - 4)*x^3 + (n - 1)*x^4)); 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 [-1, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
Related ?
Candidate equalities ?
- A249250 → x ↦ x+1 → reverse 15 rows; no all-index equality certificate was found
Same fingerprint ?
References
- [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