A182603 -id:A182603 - cp's OEIS frontend

A221578 A sum over partitions (q=6), see first comment.

1, 5, 35, 210, 1290, 7735, 46620, 279685, 1679370, 10076190, 60464670, 362787810, 2176773305, 13060638360, 78364108620, 470184650495, 2821109573550, 16926657432510, 101559954663930, 609359727929610, 3656158427989830, 21936950567886270, 131621703769781995

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=6 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..400

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*6^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 24 2013

b[n_] := Sum[EulerPhi[d]*6^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-6*x^n)  );
v=Vec(gf)

A221579 A sum over partitions (q=10), see first comment.

Original entry on oeis.org

1, 9, 99, 990, 9990, 99891, 999900, 9998901, 99998910, 999989010, 9999989010, 99999889110, 999999890109, 9999998890200, 99999998891100, 999999988901199, 9999999988902090, 99999999888912090, 999999999889011990, 9999999998889021990

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=10 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..500

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*10^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 25 2013

b[n_] := Sum[EulerPhi[d]*10^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-10*x^n)  );
v=Vec(gf)

A221580 A sum over partitions (q=12), see first comment.

Original entry on oeis.org

1, 11, 143, 1716, 20724, 248677, 2985840, 35829937, 429979836, 5159757900, 61917341772, 743008099548, 8916100178843, 106993202123808, 1283918461295184, 15407021535521759, 184884258855973380, 2218611106271412996, 26623333280416468596, 319479999364994391924

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=12 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*12^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*12^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-12*x^n)  );
v=Vec(gf)

A221581 A sum over partitions (q=14), see first comment.

Original entry on oeis.org

1, 13, 195, 2730, 38402, 537615, 7529340, 105410565, 1475786130, 20661005638, 289254613830, 4049564590890, 56693911799265, 793714765148760, 11112006817455180, 155568095444334495, 2177953337695895942, 30491346727741970070, 426878854209048054450

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=14 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*14^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*14^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-14*x^n)  );
v=Vec(gf)

A221582 A sum over partitions (q=15), see first comment.

Original entry on oeis.org

1, 14, 224, 3360, 50610, 759136, 11390400, 170855776, 2562887040, 38443305390, 576650336640, 8649755046240, 129746337080864, 1946195056159200, 29192926013193600, 437893890197853824, 6568408355529888210, 98526125332947516960, 1477891880032655307360

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=15 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*15^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*15^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-15*x^n)  );
v=Vec(gf)

A221583 A sum over partitions (q=18), see first comment.

Original entry on oeis.org

1, 17, 323, 5814, 104958, 1889227, 34011900, 612213877, 11019954438, 198359179578, 3570467115834, 64268408079198, 1156831379431973, 20822964829665048, 374813367546080412, 6746640615829343087, 121439531095946141922, 2185911559727028566514

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=18 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P=[p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
  q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
  q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
  q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
  q=6: A221578, q=10: A221579, q=12: A221580,
  q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*18^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*18^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-18*x^n)  );
v=Vec(gf)

A221584 A sum over partitions (q=20), see first comment.

Original entry on oeis.org

1, 19, 399, 7980, 159980, 3199581, 63999600, 1279991601, 25599991620, 511999832020, 10239999832020, 204799996632420, 4095999996640419, 81919999932640800, 1638399999932648400, 32767999998652808799, 655359999998652816380, 13107199999973052976380

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=20 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P=[p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
  q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
  q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
  q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
  q=6: A221578, q=10: A221579, q=12: A221580,
  q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-20*x^n)  );
v=Vec(gf)

A319753 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - k*x^j).

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 1, 3, 8, 6, 0, 1, 1, 4, 15, 24, 14, 0, 0, 1, 5, 24, 60, 78, 27, 0, 1, 1, 6, 35, 120, 252, 232, 60, 0, 0, 1, 7, 48, 210, 620, 1005, 720, 117, 0, 0, 1, 8, 63, 336, 1290, 3096, 4080, 2152, 246, 0, 0, 1, 9, 80, 504, 2394, 7735, 15600, 16305, 6528, 490, 0, 0

Offset: 0

Ilya Gutkovskiy, Sep 27 2018

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
  -1,  0,   1,    2,     3,     4,  ...
  -1,  0,   3,    8,    15,    24,  ...
   0,  0,   6,   24,    60,   120,  ...
   0,  0,  14,   78,   252,   620,  ...
   1,  0,  27,  232,  1005,  3096,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..20 give A010815, A000007, A006951, A006952, A049314, A049315, A221578, A049316, A182603, A182604, A221579, A182605, A221580, A182606, A221581, A221582, A182607, A182608, A221583, A182609, A221584.

Main diagonal gives A319754.

Table[Function[k, SeriesCoefficient[Product[(1 - x^i)/(1 - k x^i), {i, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[Sum[d (k^(i/d) - 1), {d, Divisors[i]}] x^i/i, {i, n}]], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 - x^j)/(1 - k*x^j).

G.f. of column k: exp(Sum_{j>=1} ( Sum_{d|j} d*(k^(j/d) - 1) ) * x^j/j).

cp's OEIS Frontend

A221578 A sum over partitions (q=6), see first comment.

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A221579 A sum over partitions (q=10), see first comment.

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A221580 A sum over partitions (q=12), see first comment.

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A221581 A sum over partitions (q=14), see first comment.

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A221582 A sum over partitions (q=15), see first comment.

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A221583 A sum over partitions (q=18), see first comment.

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A221584 A sum over partitions (q=20), see first comment.

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A319753 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - k*x^j).

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