A370709
a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).
Original entry on oeis.org
1, 2, 2, 20, 6, 108, 148, 776, -186, 5964, -4, 51032, -89700, 512120, -1259416, 6406032, -19733434, 78363148, -268823572, 1047941688, -3800035916, 14327505832, -52766730600, 199492430192, -746479735524, 2811936761016, -10588174502568, 40092283176560, -151796846803592
Offset: 0
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nmax = 30; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 2*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-2, x]/3], {x, 0, nmax}], x] * 2^Range[0, nmax]
A370716
a(n) = 3^(2*n) * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).
Original entry on oeis.org
1, 6, 18, 1170, -1890, 133326, 101250, 20498994, -164656314, 3778220862, -28085954094, 771567716970, -10691904063114, 183594050113518, -2711145260068326, 49416883617381354, -789899109743435994, 13176840267952166070, -216403389726994588086, 3681309971143060236810
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])
nmax = 20; CoefficientList[Series[Product[(1 + 2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[(QPochhammer[-2, x]/3)^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])
A259400
a(n) = Sum_{k=0..n} 2^k*p(k), where p(k) is the partition function A000041.
Original entry on oeis.org
1, 3, 11, 35, 115, 339, 1043, 2963, 8595, 23955, 66963, 181651, 497043, 1324435, 3536275, 9303443, 24442259, 63370643, 164296083, 421197203, 1078654355, 2739598739, 6942291347, 17469994387, 43894109587, 109593687443, 273070880147, 677066241427, 1675109266835
Offset: 0
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Table[Sum[2^k*PartitionsP[k],{k,0,n}],{n,0,40}]
A327549
Number T(n,k) of compositions of partitions of n with exactly k compositions; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 8, 8, 2, 1, 0, 16, 16, 8, 2, 1, 0, 32, 48, 24, 8, 2, 1, 0, 64, 96, 64, 24, 8, 2, 1, 0, 128, 256, 160, 80, 24, 8, 2, 1, 0, 256, 512, 448, 192, 80, 24, 8, 2, 1, 0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1
Offset: 0
T(3,1) = 4: 3, 21, 12, 111.
T(3,2) = 2: 2|1, 11|1.
T(3,3) = 1: 1|1|1.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 4, 2, 1;
0, 8, 8, 2, 1;
0, 16, 16, 8, 2, 1;
0, 32, 48, 24, 8, 2, 1;
0, 64, 96, 64, 24, 8, 2, 1;
0, 128, 256, 160, 80, 24, 8, 2, 1;
0, 256, 512, 448, 192, 80, 24, 8, 2, 1;
0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1;
...
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+expand(2^(i-1)*x*b(n-i, min(n-i, i)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);
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b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + 2^(i-1) x b[n-i, Min[n-i, i]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)
A352402
Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).
Original entry on oeis.org
1, -1, -1, -3, -1, -7, -1, -15, 31, -63, 159, -95, 671, -287, 3231, -2975, 15519, -7839, 44191, -34975, 224415, -291999, 863391, -990367, 2927775, -4902047, 12561567, -27225247, 56470687, -102640799, 152153247, -422620319, 877243551, -2278272159, 3357125791
Offset: 0
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nmax = 34; CoefficientList[Series[Product[1/(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 2^(n - k), {k, 0, n}], {n, 0, 34}]
Showing 1-5 of 5 results.
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