A249250 → basis A013609 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(14 + 122\,x + 410\,x^{2} + 698\,x^{3} + 644\,x^{4} + 308\,x^{5} + 60\,x^{6} - 26\,n - 206\,n x - 656\,n x^{2} - 1082\,n x^{3} - 980\,n x^{4} - 464\,n x^{5} - 90\,n x^{6} + 12\,n^{2} + 84\,n^{2} x + 246\,n^{2} x^{2} + 384\,n^{2} x^{3} + 336\,n^{2} x^{4} + 156\,n^{2} x^{5} + 30\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(21 + 207\,x + 838\,x^{2} + 1822\,x^{3} + 2313\,x^{4} + 1723\,x^{5} + 700\,x^{6} + 120\,x^{7} - 12\,n - 114\,n x - 448\,n x^{2} - 952\,n x^{3} - 1188\,n x^{4} - 874\,n x^{5} - 352\,n x^{6} - 60\,n x^{7}\right)\,P_{n-1}'(x) + \left(3 + 36\,x + 181\,x^{2} + 500\,x^{3} + 835\,x^{4} + 868\,x^{5} + 551\,x^{6} + 196\,x^{7} + 30\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 10\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 4 | 10 | 14 | ||||||
| 2 | 10 | 44 | 46 | 100 | |||||
| 3 | 28 | 180 | 372 | 244 | 824 | ||||
| 4 | 76 | 640 | 1968 | 2608 | 1252 | 6544 | |||
| 5 | 208 | 2168 | 8848 | 17632 | 17120 | 6472 | 52448 | ||
| 6 | 568 | 7056 | 35880 | 95456 | 139944 | 107088 | 33400 | 419392 | |
| 7 | 1552 | 22384 | 136272 | 453488 | 890096 | 1029648 | 649648 | 172432 | 3355520 |
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Technical details
- Node id
n-e82344c5ef80ab5d· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-c6176a18b14d9443
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: 4, 10; 2: 10, 44, 46; 3: 28, 180, 372, 244; 4: 76, 640, 1968, 2608, 1252; 5: 208, 2168, 8848, 17632, 17120, 6472; 6: 568, 7056, 35880, 95456, 139944, 107088, 33400; 7: 1552, 22384, 136272, 453488, 890096, 1029648, 649648, 172432; 8: 4240, 69632, 493696, 1972352, 4852960, 7526144, 7181056, 3853184, 890128; 9: 11584, 213408, 1726848, 8050944, 23821056, 46365120, 59343744, 48150528, 22470336, 4595104;
%F FormulaThe row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (12*n^2-26*n+14+(84*n^2-206*n+122)*x+(246*n^2-656*n+410)*x^2+(384*n^2-1082*n+698)*x^3+(336*n^2-980*n+644)*x^4+(156*n^2-464*n+308)*x^5+(30*n^2-90*n+60)*x^6)*P(n-1,x) - (12*n-21+(114*n-207)*x+(448*n-838)*x^2+(952*n-1822)*x^3+(1188*n-2313)*x^4+(874*n-1723)*x^5+(352*n-700)*x^6+(60*n-120)*x^7)*d/dx P(n-1,x) + (3+36*x+181*x^2+500*x^3+835*x^4+868*x^5+551*x^6+196*x^7+30*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 10*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 4 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 26*n + 14 + (84*n^2 - 206*n + 122)*x + (246*n^2 - 656*n + 410)*x^2 + (384*n^2 - 1082*n + 698)*x^3 + (336*n^2 - 980*n + 644)*x^4 + (156*n^2 - 464*n + 308)*x^5 + (30*n^2 - 90*n + 60)*x^6)*P[n - 1, x] - (12*n - 21 + (114*n - 207)*x + (448*n - 838)*x^2 + (952*n - 1822)*x^3 + (1188*n - 2313)*x^4 + (874*n - 1723)*x^5 + (352*n - 700)*x^6 + (60*n - 120)*x^7)*D[P[n - 1, x], x] + (3 + 36*x + 181*x^2 + 500*x^3 + 835*x^4 + 868*x^5 + 551*x^6 + 196*x^7 + 30*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 26*n + 14 + (84*n^2 - 206*n + 122)*x + (246*n^2 - 656*n + 410)*x^2 + (384*n^2 - 1082*n + 698)*x^3 + (336*n^2 - 980*n + 644)*x^4 + (156*n^2 - 464*n + 308)*x^5 + (30*n^2 - 90*n + 60)*x^6)*v[i - 1] - (12*n - 21 + (114*n - 207)*x + (448*n - 838)*x^2 + (952*n - 1822)*x^3 + (1188*n - 2313)*x^4 + (874*n - 1723)*x^5 + (352*n - 700)*x^6 + (60*n - 120)*x^7)*deriv(v[i - 1], x) + (3 + 36*x + 181*x^2 + 500*x^3 + 835*x^4 + 868*x^5 + 551*x^6 + 196*x^7 + 30*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); 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
basis_map_A013609 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Geometric basis map (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.?
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from
basis_map_A013609 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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