A319110
Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).
Original entry on oeis.org
1, 0, 1, 2, 4, 6, 13, 18, 37, 56, 101, 152, 285, 410, 713, 1118, 1830, 2780, 4618, 6934, 11278, 17092, 26894, 40822, 64435, 96372, 149299, 225104, 345131, 515394, 788176, 1169962, 1772957, 2632458, 3950365, 5849260, 8748993, 12867848, 19135894, 28126614, 41598695
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0 or i=1,
0^n, b(n, i-1)+(i-1)*b(n-i, min(n-i, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..42); # Alois P. Heinz, Aug 19 2019
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nmax = 40; CoefficientList[Series[Product[1/(1 - (k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[(j - 1)^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d - 1)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]
A339426
Number of compositions (ordered partitions) of n into an even number of powers of 2.
Original entry on oeis.org
1, 0, 1, 2, 2, 6, 9, 14, 30, 48, 86, 156, 268, 478, 849, 1486, 2638, 4660, 8214, 14532, 25664, 45304, 80078, 141412, 249768, 441276, 779376, 1376696, 2431924, 4295534, 7587753, 13403102, 23674870, 41819588, 73870046, 130483396, 230486384, 407130332, 719153726
Offset: 0
a(5) = 6 because we have [4, 1], [1, 4], [2, 1, 1, 1], [1, 2, 1, 1], [1, 1, 2, 1] and [1, 1, 1, 2].
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b:= proc(n, t) option remember; `if`(n=0, t,
add(b(n-2^i, 1-t), i=0..ilog2(n)))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..42); # Alois P. Heinz, Dec 03 2020
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nmax = 38; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]) + 1/(1 + Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}])), {x, 0, nmax}], x]
A271176
Expansion of -(4*x^3-7*x^2+4*x-1)/(2*x^4-5*x^3+8*x^2-5*x+1).
Original entry on oeis.org
1, 1, 4, 13, 36, 94, 239, 597, 1471, 3586, 8669, 20818, 49726, 118259, 280239, 662117, 1560516, 3670321, 8617584, 20203698, 47308391, 110659649, 258614439, 603929562, 1409413761, 3287385206, 7664034874, 17860302403
Offset: 0
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Table[(n + 1) Sum[Sum[(Binomial[i + k, i] 2^i Binomial[2 k + 2, n - i - k] (-1)^(n - i - k))/(k + 1), {i, 0, n - k}], {k, 0, n}], {n, 0, 27}] (* or *)
CoefficientList[Series[-(4 x^3 - 7 x^2 + 4 x - 1)/(2 x^4 - 5 x^3 + 8 x^2 - 5 x + 1), {x, 0, 27}], x] (* Michael De Vlieger, Apr 01 2016 *)
LinearRecurrence[{5,-8,5,-2},{1,1,4,13},30] (* Harvey P. Dale, Jan 19 2021 *)
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a(n):=(n+1)*sum(sum(binomial(i+k,i)*2^i*binomial(2*k+2,n-i-k)*(-1)^(n-i-k),i,0,n-k)/(k+1),k,0,n);
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x='x+O('x^99); Vec(-(4*x^3-7*x^2+4*x-1)/(2*x^4-5*x^3+8*x^2-5*x+1)) \\ Altug Alkan, Apr 01 2016
A271180
Expansion of (4*x^3-7*x^2+4*x-1)/(x^6-4*x^5+4*x^4+x^3-7*x^2+5*x-1).
Original entry on oeis.org
1, 1, 5, 15, 45, 125, 342, 921, 2461, 6535, 17282, 45567, 119898, 315020, 826830, 2168583, 5684731, 14896459, 39024899, 102216045, 267693813, 700997144, 1835543565, 4806092673, 12583591525, 32946281848, 86258240735, 225834015840
Offset: 0
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Table[(n + 1) Sum[Sum[(Binomial[i + k, i] 2^i Binomial[2 k + 2, n - i - k] (-1)^(n - i - k))/(k + 1) Fibonacci[k + 1], {i, 0, n - k}], {k, 0, n}], {n, 0, 27}] (* or *)
CoefficientList[Series[(4 x^3 - 7 x^2 + 4 x - 1)/(x^6 - 4 x^5 + 4 x^4 + x^3 - 7 x^2 + 5 x - 1), {x, 0, 27}], x] (* Michael De Vlieger, Apr 01 2016 *)
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a(n):=(n+1)*sum(sum(binomial(i+k,i)*2^i*binomial(2*k+2,n-i-k)*(-1)^(n-i-k),i,0,n-k)/(k+1)*fib(k+1),k,0,n);
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x='x+O('x^99); Vec((4*x^3-7*x^2+4*x-1)/(x^6-4*x^5+4*x^4+x^3-7*x^2+5*x-1)) \\ Altug Alkan, Apr 01 2016
