A293806
a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k)).
Original entry on oeis.org
1, 1, 1, 4, 6, 8, 11, 14, 19, 24, 30, 37, 47, 57, 70, 84, 102, 121, 144, 170, 202, 235, 275, 319, 372, 429, 495, 567, 652, 742, 848, 963, 1095, 1237, 1399, 1574, 1775, 1990, 2235, 2499, 2795, 3114, 3473, 3859, 4292, 4755, 5271, 5827, 6444, 7107, 7840, 8625, 9493, 10422, 11444, 12541
Offset: 0
a(4) = 6 because we have [4], [1a, 1a, 1a, 1a], [1a, 1a, 1a, 1b], [1a, 1a, 1b, 1b], [1a, 1b, 1b, 1b] and [1b, 1b, 1b, 1b].
G.f. = -x - 2*x^2 + 1/((1 - x)*(1 - x)*(1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^11)*(1 - x^14)*(1 - x^19)*...) = 1 + x + x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 14*x^7 + 19*x^8 + ...
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(a(i)>n, 0, b(n-a(i), i))))
end:
a:= n-> `if`(n<2, 1, b(n, n-1)):
seq(a(n), n=0..60); # Alois P. Heinz, Oct 16 2017
-
a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 55}]
A293807
a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k))^a(k).
Original entry on oeis.org
1, 1, 1, 4, 9, 14, 19, 24, 39, 63, 87, 111, 155, 235, 329, 423, 552, 771, 1091, 1430, 1825, 2400, 3295, 4392, 5597, 7117, 9367, 12476, 16077, 20182, 25677, 33472, 43406, 54578, 68109, 86475, 111316, 140965, 174836, 217520, 275130, 348555, 433578, 533640, 662620, 831747
Offset: 0
a(3) = 4 because we have [1a, 1a, 1a], [1a, 1a, 1b], [1a, 1b, 1b] and [1b, 1b, 1b].
G.f. = -x - 2*x^2 + 1/((1 - x)*(1 - x)*(1 - x^4)^4*(1 - x^9)^9*(1 - x^14)^14*(1 - x^19)^19*(1 - x^24)^24*(1 - x^39)^39*...) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 14*x^5 + 19*x^6 + 24*x^7 + 39*x^8 + ...
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-a(i)*j, i-1)*binomial(a(i)+j-1, j), j=0..n/a(i))))
end:
a:= n-> `if`(n<2, 1, b(n, n-1)):
seq(a(n), n=0..60); # Alois P. Heinz, Oct 16 2017
-
a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^a[k])^a[k], {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 45}]
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from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==0 else 0 if i<1 else sum(b(n - a(i)*j, i - 1) * binomial(a(i) + j - 1, j) for j in range(n//a(i) + 1))
def a(n): return 1 if n<2 else b(n, n - 1)
print([a(n) for n in range(51)]) # Indranil Ghosh, Dec 13 2017, after Maple code
A296387
a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} (1 + x^a(k))/(1 - x^a(k)).
Original entry on oeis.org
1, 1, 2, 6, 8, 10, 14, 18, 26, 34, 46, 58, 74, 90, 114, 138, 174, 210, 260, 310, 378, 446, 536, 626, 748, 870, 1034, 1198, 1410, 1622, 1892, 2162, 2510, 2858, 3306, 3754, 4316, 4878, 5576, 6274, 7144, 8014, 9096, 10178, 11508, 12838, 14458, 16078, 18048, 20018, 22410, 24802, 27690, 30578, 34040
Offset: 0
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a[n_] := a[n] = SeriesCoefficient[Product[(1 + x^a[k])/(1 - x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 54}]
A300411
a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - a(k)*x^a(k)).
Original entry on oeis.org
1, 1, 1, 4, 9, 14, 19, 24, 45, 75, 105, 135, 229, 359, 503, 647, 1047, 1591, 2272, 2972, 4696, 6996, 9844, 12894, 20064, 29538, 41204, 54407, 84457, 123723, 171757, 225939, 348643, 508693, 703815, 923529, 1423892, 2076942, 2870977, 3763380, 5778379, 8414332, 11621307
Offset: 0
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a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - a[k] x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 42}]
A307545
G.f. A(x) satisfies: A(x) = 1/(1 - x^a(0) - x^a(1) - x^a(2) - x^a(3) - ...) - x.
Original entry on oeis.org
1, 1, 4, 8, 17, 36, 76, 160, 338, 714, 1508, 3184, 6723, 14196, 29976, 63296, 133653, 282217, 595920, 1258324, 2657032, 5610494, 11846920, 25015536, 52821917, 111536883, 235517320, 497310008, 1050102149, 2217358398, 4682095232, 9886545984, 20876079330, 44081187594, 93080269957
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 17*x^4 + 36*x^5 + 76*x^6 + 160*x^7 + 338*x^8 + ... = 1/(1 - 2*x - x^4 - x^8 - x^17 - x^36 - x^76 - x^160 - x^338 - ...) - x.
A339246
a(n) = 1 for n <= 2; thereafter a(n) is the number of partitions of n where every part k appears at least a(k) times.
Original entry on oeis.org
1, 1, 1, 2, 3, 3, 5, 5, 7, 8, 10, 11, 15, 16, 20, 23, 28, 31, 38, 41, 50, 56, 66, 72, 86, 94, 110, 122, 140, 154, 178, 195, 223, 245, 276, 303, 344, 376, 421, 461, 513, 561, 627, 681, 756, 824, 909, 988, 1092, 1182, 1301, 1413, 1547, 1673, 1834, 1979, 2165, 2341, 2548, 2746, 2993
Offset: 0
a(0) = a(1) = a(2) = 1: by definition.
a(3) = 2: [2, 1], [1, 1, 1].
a(4) = 3: [2, 2], [2, 1, 1], [1, 1, 1, 1].
a(5) = 3: [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1].
a(6) = 5: [3, 3], [2, 2, 2], [2, 2, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1].
Showing 1-6 of 6 results.
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