A240740
Number of compositions of n having exactly five fixed points.
Original entry on oeis.org
1, 1, 3, 7, 16, 35, 70, 155, 321, 665, 1368, 2802, 5711, 11623, 23526, 47567, 95967, 193316, 388893, 781519, 1569154, 3148292, 6313052, 12652917, 25349663, 50770869, 101658425, 203506976, 407323589, 815151106, 1631122032, 3263576647, 6529319168, 13062156519
Offset: 15
a(17) = 3: 123416, 123452, 1234511.
a(18) = 7: 123156, 123426, 123453, 1234161, 1234512, 1234521, 12345111.
a(19) = 16: 121456, 123256, 123436, 123454, 1231561, 1234117, 1234162, 1234261, 1234513, 1234522, 1234531, 12341611, 12345112, 12345121, 12345211, 123451111.
-
b:= proc(n, i) option remember; `if`(n=0, 1, series(
add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 6))
end:
a:= n-> coeff(b(n, 1), x, 5):
seq(a(n), n=15..50);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 6}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 5}]; Table[a[n], {n, 15, 50}] (* Jean-François Alcover, Nov 07 2014, after Maple *)
A240741
Number of compositions of n having exactly six fixed points.
Original entry on oeis.org
1, 1, 3, 7, 16, 35, 76, 155, 334, 691, 1427, 2928, 5985, 12181, 24718, 50052, 101060, 203767, 410240, 824943, 1657225, 3326530, 6672880, 13378262, 26809661, 53706442, 107555030, 215342201, 431063039, 862743300, 1726491928, 3454620480, 6911903675, 13828137410
Offset: 21
-
b:= proc(n, i) option remember; `if`(n=0, 1, series(
add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 7))
end:
a:= n-> coeff(b(n, 1), x, 6):
seq(a(n), n=21..60);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 7}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 6}]; Table[a[n], {n, 21, 60}] (* Jean-François Alcover, Nov 07 2014, after Maple *)
A240742
Number of compositions of n having exactly seven fixed points.
Original entry on oeis.org
1, 1, 3, 7, 16, 35, 76, 162, 334, 706, 1457, 2996, 6130, 12496, 25383, 51423, 103937, 209723, 422418, 849843, 1707917, 3429407, 6881120, 13798903, 27657921, 55414350, 110989891, 222243203, 444916908, 890536103, 1782217460, 3566301121, 7135641348, 14276228900
Offset: 28
-
b:= proc(n, i) option remember; `if`(n=0, 1, series(
add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 8))
end:
a:= n-> coeff(b(n, 1), x, 7):
seq(a(n), n=28..65);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n - j, i + 1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 8}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 7}]; Table[a[n], {n, 28, 65}] (* Jean-François Alcover, Nov 06 2014, after Maple *)
A240743
Number of compositions of n having exactly eight fixed points.
Original entry on oeis.org
1, 1, 3, 7, 16, 35, 76, 162, 342, 706, 1474, 3030, 6207, 12660, 25739, 52174, 105516, 212972, 429169, 863721, 1736237, 3487091, 6998235, 14036039, 28137051, 56380699, 112936022, 226157834, 452782897, 906328973, 1813903281, 3629837847, 7262985540, 14531361628
Offset: 36
-
b:= proc(n, i) option remember; `if`(n=0, 1, series(
add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 9))
end:
a:= n-> coeff(b(n, 1), x, 8):
seq(a(n), n=36..70);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n - j, i + 1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 9}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 8}]; Table[a[n], {n, 36, 70}] (* Jean-François Alcover, Nov 06 2014, after Maple *)
A240744
Number of compositions of n having exactly nine fixed points.
Original entry on oeis.org
1, 1, 3, 7, 16, 35, 76, 162, 342, 715, 1474, 3049, 6245, 12746, 25922, 52571, 106353, 214731, 432827, 871240, 1751725, 3518787, 7062725, 14167004, 28402284, 56916681, 114017164, 228335406, 457163368, 915131854, 1831578490, 3665302380, 7334102844, 14673905376
Offset: 45
-
b:= proc(n, i) option remember; `if`(n=0, 1, series(
add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 10))
end:
a:= n-> coeff(b(n, 1), x, 9):
seq(a(n), n=45..80);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n - j, i + 1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 10}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 9}]; Table[a[n], {n, 45, 80}] (* Jean-François Alcover, Nov 06 2014, after Maple *)
A240745
Number of compositions of n having exactly ten fixed points.
Original entry on oeis.org
1, 1, 3, 7, 16, 35, 76, 162, 342, 715, 1484, 3049, 6266, 12788, 26017, 52773, 106791, 215654, 434766, 875271, 1760058, 3535850, 7097682, 14238236, 28546852, 57209494, 114608933, 229529157, 459567874, 919969090, 1841299703, 3684822059, 7373269689, 14752449235
Offset: 55
-
b:= proc(n, i) option remember; `if`(n=0, 1, series(
add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 11))
end:
a:= n-> coeff(b(n, 1), x, 10):
seq(a(n), n=55..90);
-
$RecursionLimit = 500; b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n - j, i + 1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 11}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 10}]; Table[a[n], {n, 55, 90}] (* Jean-François Alcover, Nov 06 2014, after Maple *)
A383175
Number of compositions of n such that any fixed point k can be k different colors.
Original entry on oeis.org
1, 1, 2, 5, 10, 22, 48, 101, 213, 450, 945, 1961, 4064, 8385, 17242, 35332, 72141, 146924, 298552, 605377, 1225277, 2475912, 4995754, 10067848, 20267680, 40762951, 81916919, 164504411, 330155437, 662265817, 1327860471, 2661376529, 5332341881, 10680912173
Offset: 0
a(3) = 5 counts: (3), (2,1), (1_a,2_a), (1_a,2_b), (1_a,1,1).
-
b:= proc(n, i) option remember; `if`(n=0, 1, add(
`if`(n<=i+j, ceil(2^(n-j-1)), b(n-j, i+1))*
`if`(i=j, j, 1), j=1..n))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..33); # Alois P. Heinz, Apr 18 2025
-
A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N, prod(j=1,i, j*x^j-x^j+x/(1-x))))}
A_x(30)