A035607 → basis A013609 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - 5\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(41 + 39\,x + 8\,x^{2} - 14\,n - 12\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(21 + 32\,x + 15\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 7 | 4 | 11 | |||||
| 3 | 31 | 32 | 8 | 71 | ||||
| 4 | 145 | 216 | 104 | 16 | 481 | |||
| 5 | 673 | 1312 | 936 | 288 | 32 | 3241 | ||
| 6 | 3127 | 7532 | 7112 | 3280 | 736 | 64 | 21851 | |
| 7 | 14527 | 41664 | 48944 | 30080 | 10176 | 1792 | 128 | 147311 |
Compute more
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Technical details
- Node id
n-9681de120ff6e17e· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-1bf4f6258e36ae6b
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 7, 4; 3: 31, 32, 8; 4: 145, 216, 104, 16; 5: 673, 1312, 936, 288, 32; 6: 3127, 7532, 7112, 3280, 736, 64; 7: 14527, 41664, 48944, 30080, 10176, 1792, 128; 8: 67489, 224560, 315504, 242272, 109632, 29184, 4224, 256; 9: 313537, 1187264, 1941296, 1788224, 1013760, 361728, 79232, 9728, 512;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (2*n^2-5*n+2)*P(n-1,x) - (14*n-41+(12*n-39)*x+(2*n-8)*x^2)*d/dx P(n-1,x) + (21+32*x+15*x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 7 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2)*P[n - 1, x] - (14*n - 41 + (12*n - 39)*x + (2*n - 8)*x^2)*D[P[n - 1, x], x] + (21 + 32*x + 15*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}])/(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; v[3] = 7 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2)*v[i - 1] - (14*n - 41 + (12*n - 39)*x + (2*n - 8)*x^2)*deriv(v[i - 1], x) + (21 + 32*x + 15*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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 (−∞, -1]
- 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 Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
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Geometric basis map (elementary) polytool ? step ›
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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from
basis_map_A013609 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
Related ?
Candidate equalities ?
- A035607 → basis A013609 → basis A007318 15 rows; no all-index equality certificate was found
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