A341390 -id:A341390 - cp's OEIS frontend

A341384 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^2.

1, 4, 14, 36, 89, 200, 434, 898, 1810, 3548, 6810, 12816, 23719, 43250, 77795, 138244, 242920, 422510, 727907, 1243094, 2105493, 3538936, 5905481, 9787810, 16118588, 26383244, 42936039, 69491436, 111884015, 179239648, 285775148, 453550910, 716670609

Offset: 2

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A026007, A026011, A321947, A327380, A341385, A341386, A341387, A341388, A341390, A341391.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..34);  # Alois P. Heinz, Feb 10 2021

nmax = 34; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &

a(n) ~ A026011(n). - Vaclav Kotesovec, Feb 20 2021

A341385 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^3.

Original entry on oeis.org

1, 6, 27, 92, 279, 762, 1952, 4725, 10968, 24551, 53346, 112932, 233755, 474288, 945384, 1854517, 3585534, 6841182, 12895246, 24035841, 44337672, 80999765, 146644746, 263249169, 468817933, 828658233, 1454315508, 2535217624, 4391290854, 7560034419, 12939963016

Offset: 3

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Cf. A026007, A027346, A321948, A327381, A341384, A341386, A341387, A341388, A341390, A341391.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

a(n) ~ A027346(n). - Vaclav Kotesovec, Feb 20 2021

A341386 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^4.

Original entry on oeis.org

1, 8, 44, 184, 662, 2120, 6256, 17276, 45277, 113568, 274592, 643220, 1465838, 3260428, 7097338, 15153288, 31791822, 65645360, 133584864, 268213400, 531879490, 1042657088, 2022113788, 3882468712, 7384455791, 13921287616, 26026092198, 48273051172, 88868177735

Offset: 4

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A026007, A027906, A321949, A327382, A341384, A341385, A341387, A341388, A341390, A341391.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..32);  # Alois P. Heinz, Feb 10 2021

nmax = 32; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

A341387 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^5.

Original entry on oeis.org

1, 10, 65, 320, 1330, 4872, 16255, 50335, 146775, 407045, 1082000, 2773045, 6884650, 16620225, 39135280, 90113553, 203347645, 450516450, 981491380, 2105504205, 4452798556, 9293254605, 19158353285, 39044262235, 78719105560, 157112112293, 310599279105

Offset: 5

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A026007, A321950, A327383, A341384, A341385, A341386, A341388, A341390, A341391.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..31);  # Alois P. Heinz, Feb 10 2021

nmax = 31; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

A341388 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^6.

Original entry on oeis.org

1, 12, 90, 508, 2391, 9840, 36578, 125358, 402093, 1220232, 3532836, 9821280, 26352110, 68528718, 173311971, 427486178, 1030855416, 2435255634, 5645810201, 12864839166, 28850671284, 63751119334, 138946592610, 298974483954, 635626314025

Offset: 6

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Cf. A026007, A321951, A327384, A341384, A341385, A341386, A341387, A341390, A341391.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..30);  # Alois P. Heinz, Feb 10 2021

nmax = 30; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

A341391 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^8.

Original entry on oeis.org

1, 16, 152, 1072, 6204, 31024, 138544, 564824, 2135902, 7580944, 25485560, 81734696, 251514840, 746123304, 2142114356, 5971477112, 16208165181, 42936937488, 111240873128, 282363615336, 703303327288, 1721329848680, 4144792701532, 9829483710112

Offset: 8

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Cf. A026007, A321953, A327386, A341384, A341385, A341386, A341387, A341388, A341390.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..31);  # Alois P. Heinz, Feb 10 2021

nmax = 31; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

A341393 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^9.

Original entry on oeis.org

1, 18, 189, 1464, 9252, 50292, 243117, 1068939, 4344660, 16522967, 59349627, 202844007, 663615180, 2088375867, 6347592999, 18698498610, 53538715836, 149375490453, 406987481852, 1084906793142, 2834211905622, 7266665613438, 18308976116535

Offset: 9

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Cf. A026007, A321954, A327387, A341384, A341385, A341386, A341387, A341388, A341390, A341391, A341394.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..31);  # Alois P. Heinz, Feb 10 2021

nmax = 31; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

A341394 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^10.

Original entry on oeis.org

1, 20, 230, 1940, 13285, 77944, 405250, 1910330, 8300380, 33655860, 128574734, 466317760, 1615509765, 5373215450, 17230062315, 53457917856, 160963157005, 471587847690, 1347417640405, 3761860656610, 10280578499844, 27543107112940, 72440412567485

Offset: 10

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..10000

Cf. A026007, A321955, A327388, A341384, A341385, A341386, A341387, A341388, A341390, A341391, A341393.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..32);  # Alois P. Heinz, Feb 10 2021

nmax = 32; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

A341395 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.

Original entry on oeis.org

1, 2, 14, 92, 662, 4872, 36578, 278161, 2135902, 16522967, 128574734, 1005321616, 7891885382, 62160038813, 491003317483, 3888045701232, 30854283708670, 245315312649653, 1953735732991919, 15583347966328833, 124463844976490422, 995305632560023009, 7968042676400949882

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1097

Cf. A026007, A257675, A270913, A270922, A324595, A341384, A341385, A341386, A341387, A341388, A341390, A341391, A341393, A341394.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, g(n+1), (q->
      add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 10 2021

Join[{1}, Table[SeriesCoefficient[(-1 + Product[(1 + x^k)^k, {k, 1, 2 n}])^n, {x, 0, 2 n}], {n, 1, 22}]]
A[n_, k_] := A[n, k] = If[n == 0, 1, k Sum[A[n - j, k] Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 22}]

a(n) ~ c * d^n / sqrt(n), where d = 8.191928734348241613884260036383361206707761707495484130816183793791732456844... and c = 0.30227512720649344220720362916140286571342247518684432176920275576011986255... - Vaclav Kotesovec, Feb 20 2021

cp's OEIS Frontend

A341384 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^2.

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A341385 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^3.

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A341386 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^4.

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A341387 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^5.

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A341388 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^6.

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A341391 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^8.

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A341393 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^9.

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Maple

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A341394 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^10.

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Maple

Mathematica

A341395 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.

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