A185646
Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))).
Original entry on oeis.org
1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 3, 3, 1, 0, 1, 1, 1, 2, 3, 4, 5, 1, -1, 1, 1, 1, 2, 3, 5, 7, 6, 1, 0, 1, 1, 1, 2, 3, 5, 8, 11, 10, 1, 0, 1, 1, 1, 2, 3, 5, 9, 13, 17, 14, 1, 0, 1, 1, 1, 2, 3, 5, 9, 14, 22, 28, 21, 1, 0
Offset: 0
Square array A(n,m) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, ...
0, 1, 2, 3, 3, 3, 3, 3, 3, ...
-1, 1, 3, 4, 5, 5, 5, 5, 5, ...
0, 1, 5, 7, 8, 9, 9, 9, 9, ...
0, 1, 6, 11, 13, 14, 15, 15, 15, ...
-1, 1, 10, 17, 22, 24, 25, 26, 26, ...
Columns m=0-10 give:
A143064,
A000012,
A227360,
A173173(n+1),
A227374,
A227375,
A228646,
A228644,
A185648,
A228645,
A185649.
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nMax = 12; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A = Table[col[m][[1 ;; nMax + 1]], {m, 0, nMax}] // Transpose; a[n_ /; 0 <= n <= nMax, m_ /; 0 <= m <= nMax] := With[{n1 = n + 1, m1 = m + 1}, A[[n1, m1]]]; Table[a[n - m, m], {n, 0, nMax}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2016 *)
A228644
Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=7.
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 9, 15, 26, 44, 76, 131, 225, 389, 670, 1156, 1994, 3439, 5934, 10236, 17661, 30470, 52569, 90699, 156483, 269985, 465811, 803677, 1386609, 2392357, 4127611, 7121498, 12286951, 21199078, 36575462, 63104849, 108876873, 187848862, 324101847
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,0,-1,-1,-3,-2,-1,0,2,2,3,3,1,0,0,-2,-1,-1,-1).
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a:= n-> coeff(series(-(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)), x, n+1), x, n): seq(a(n), n=0..50);
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nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228644 = col[7][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)
A228645
Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=9.
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 134, 232, 402, 695, 1205, 2086, 3613, 6259, 10841, 18780, 32531, 56354, 97621, 169111, 292954, 507488, 879136, 1522947, 2638242, 4570298, 7917253, 13715281, 23759370, 41159039, 71300984, 123516755, 213971647, 370669282
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
- Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 0, 0, -1, -1, -2, -1, -3, -2, 0, 1, 3, 4, 4, 4, 4, 2, 0, -2, -3, -5, -4, -4, -3, -2, 0, 1, 1, 2, 2, 1, 1, 1).
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a:= n-> coeff(series(-(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)), x, n+1), x, n): seq(a(n), n=0..50);
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nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A228645 = col[9][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)
Showing 1-3 of 3 results.