A035607 → basis A013611 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(4 - 10\,n + 4\,n^{2}\right)\,P_{n-1}(x) + \left(88 + 110\,x + 32\,x^{2} - 28\,n - 32\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(35 + 68\,x + 42\,x^{2} + 8\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 11 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 11 | 8 | 19 | |||||
| 3 | 71 | 96 | 32 | 199 | ||||
| 4 | 481 | 944 | 608 | 128 | 2161 | |||
| 5 | 3241 | 8352 | 7968 | 3328 | 512 | 23401 | ||
| 6 | 21851 | 69720 | 88032 | 54912 | 16896 | 2048 | 253459 | |
| 7 | 147311 | 560448 | 880320 | 730112 | 336896 | 81920 | 8192 | 2745199 |
Compute more
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Technical details
- Node id
n-d6c33c44f3c36245· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-96a3b8d1e1453386
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: 11, 8; 3: 71, 96, 32; 4: 481, 944, 608, 128; 5: 3241, 8352, 7968, 3328, 512; 6: 21851, 69720, 88032, 54912, 16896, 2048; 7: 147311, 560448, 880320, 730112, 336896, 81920, 8192; 8: 993121, 4387936, 8242752, 8528640, 5245952, 1916928, 385024, 32768; 9: 6695281, 33691584, 73651648, 91314688, 70195200, 34242560, 10346496, 1769472, 131072;
%F FormulaThe row polynomials satisfy (2*n-4)*P(n,x) = (4*n^2-10*n+4)*P(n-1,x) - (28*n-88+(32*n-110)*x+(8*n-32)*x^2)*d/dx P(n-1,x) + (35+68*x+42*x^2+8*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) = 11 + 8*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 11 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 4)*P[n - 1, x] - (28*n - 88 + (32*n - 110)*x + (8*n - 32)*x^2)*D[P[n - 1, x], x] + (35 + 68*x + 42*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 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] = 1; v[3] = 11 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = ((4*n^2 - 10*n + 4)*v[i - 1] - (28*n - 88 + (32*n - 110)*x + (8*n - 32)*x^2)*deriv(v[i - 1], x) + (35 + 68*x + 42*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n - 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]
- 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_A013611 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_A013611 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 A013611 → 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