A341245 -id:A341245 - cp's OEIS frontend

A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

1, 0, 4, 4, 10, 16, 26, 44, 63, 100, 144, 212, 297, 420, 584, 796, 1081, 1452, 1940, 2556, 3355, 4372, 5668, 7288, 9327, 11892, 15076, 19012, 23884, 29904, 37276, 46284, 57276, 70680, 86918, 106528, 130220, 158784, 193054, 234076, 283178, 341824, 411616, 494512, 592933

Offset: 4

Ilya Gutkovskiy, Feb 07 2021

nonn

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

Cf. A000700, A001482, A022599, A112160, A327382, A338463, A341222, A341241, A341244, A341245, A341246, A341247, A341251.

Column k=4 of A341279.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, 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..48);  # Alois P. Heinz, Feb 07 2021

nmax = 48; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^4.

a(n) ~ A112160(n). - Vaclav Kotesovec, Feb 20 2021

A341251 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^9.

Original entry on oeis.org

1, 0, 9, 9, 45, 81, 201, 414, 828, 1650, 3060, 5697, 10131, 17829, 30564, 51543, 85482, 139455, 224527, 356436, 559341, 867405, 1331208, 2022525, 3044331, 4542174, 6720705, 9866794, 14377941, 20804994, 29903823, 42709860, 60631011, 85575855, 120118500, 167716548

Offset: 9

Ilya Gutkovskiy, Feb 07 2021

nonn

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

Cf. A000700, A001487, A022604, A327387, A338463, A341228, A341241, A341243, A341244, A341245, A341246, A341247.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, 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..44);  # Alois P. Heinz, Feb 07 2021

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^9.

A341241 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^3.

Original entry on oeis.org

1, 0, 3, 3, 6, 9, 13, 21, 27, 40, 54, 75, 97, 129, 171, 220, 282, 360, 460, 576, 720, 896, 1116, 1374, 1682, 2061, 2517, 3050, 3684, 4449, 5354, 6414, 7656, 9135, 10875, 12891, 15243, 18015, 21243, 24966, 29286, 34326, 40156, 46851, 54573, 63509, 73794, 85551, 99035, 114555

Offset: 3

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A022598, A047655, A107635, A327381, A338463, A341221, A341243, A341244, A341245, A341246, A341247, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, 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..52);  # Alois P. Heinz, Feb 07 2021

nmax = 52; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^3.

a(n) ~ A107635(n). - Vaclav Kotesovec, Feb 20 2021

A341244 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^5.

Original entry on oeis.org

1, 0, 5, 5, 15, 25, 45, 80, 125, 210, 321, 500, 745, 1110, 1620, 2326, 3315, 4660, 6500, 8955, 12261, 16640, 22425, 29990, 39870, 52701, 69230, 90460, 117620, 152225, 196066, 251455, 321195, 408710, 518060, 654317, 823690, 1033535, 1292690, 1611970, 2004462, 2485605

Offset: 5

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001483, A022600, A327383, A338463, A341223, A341241, A341243, A341245, A341246, A341247, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, 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..46);  # Alois P. Heinz, Feb 07 2021

nmax = 46; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^5.

A341246 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^7.

Original entry on oeis.org

1, 0, 7, 7, 28, 49, 105, 203, 364, 672, 1141, 1960, 3220, 5250, 8359, 13104, 20272, 30877, 46522, 69195, 101941, 148604, 214697, 307475, 436849, 615965, 862246, 1199009, 1656642, 2275231, 3106824, 4219502, 5701066, 7664923, 10256771, 13663574, 18123924, 23941190

Offset: 7

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001485, A022602, A327385, A338463, A341226, A341241, A341243, A341244, A341245, A341247, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..44);  # Alois P. Heinz, Feb 07 2021

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^7.

A341247 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^8.

Original entry on oeis.org

1, 0, 8, 8, 36, 64, 148, 296, 562, 1080, 1920, 3440, 5890, 9992, 16532, 26920, 43175, 68144, 106260, 163472, 248824, 374504, 558212, 824208, 1206409, 1751360, 2522692, 3607456, 5122848, 7227392, 10132948, 14123000, 19573393, 26981768, 37003700, 50499952, 68595956

Offset: 8

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001486, A007259, A101127, A327386, A338463, A341227, A341241, A341243, A341244, A341245, A341246, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, 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..44);  # Alois P. Heinz, Feb 07 2021

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^8.

A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 3, 3, 4, 0, 1, 0, 1, 4, 6, 4, 5, 0, 1, 0, 2, 5, 9, 10, 5, 6, 0, 1, 0, 2, 8, 13, 16, 15, 6, 7, 0, 1, 0, 2, 9, 21, 26, 25, 21, 7, 8, 0, 1, 0, 2, 12, 27, 44, 45, 36, 28, 8, 9, 0, 1, 0, 3, 15, 40, 63, 80, 71, 49, 36, 9, 10, 0, 1

Offset: 0

Ilya Gutkovskiy, Feb 08 2021

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  0,  1;
  0,  1,  0,  1;
  0,  1,  2,  0,  1;
  0,  1,  2,  3,  0,  1;
  0,  1,  3,  3,  4,  0,  1;
  0,  1,  4,  6,  4,  5,  0,  1;
  0,  2,  5,  9, 10,  5,  6,  0,  1;
  0,  2,  8, 13, 16, 15,  6,  7,  0,  1;
  0,  2,  9, 21, 26, 25, 21,  7,  8,  0,  1;
  0,  2, 12, 27, 44, 45, 36, 28,  8,  9,  0,  1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give A000007, A000700, A338463, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253.
Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.
Row sums give A307058.
T(2n,n) gives A341265.
Cf. A000009, A047265, A060642, A286352, A308680.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 09 2021

T[n_, k_] := SeriesCoefficient[(-1 + 2/QPochhammer[-1, -x])^k, {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

G.f. of column k: (-1 + Product_{j>=1} (1 + x^(2*j-1)))^k.

Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).

A341265 Coefficient of x^(2*n) in (-1 + Product_{k>=1} 1 / (1 + x^k))^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 25, 71, 203, 562, 1650, 4667, 13673, 39427, 115440, 336639, 987628, 2898658, 8529257, 25134200, 74173606, 219207815, 648546314, 1921045953, 5695642513, 16902924883, 50203798050, 149229323544, 443895849894, 1321292939459, 3935377071154, 11728037768186

Offset: 0

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000700, A081362, A255526, A324595, A338463, A340987, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253, A341263, A341279.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, 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..30);  # Alois P. Heinz, Feb 07 2021

Table[SeriesCoefficient[(-1 + 1/QPochhammer[-x, x])^n, {x, 0, 2 n}], {n, 0, 30}]
A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] Sum[Mod[d, 2] d, {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, 30}]

a(n) = A341279(2n,n).

a(n) ~ c * d^n / sqrt(n), where d = 3.03044218957412050685579849718626198523346... and c = 0.2319377657497495246637662111041144... - Vaclav Kotesovec, Feb 20 2021

A341253 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^10.

Original entry on oeis.org

1, 0, 10, 10, 55, 100, 265, 560, 1175, 2420, 4667, 9000, 16575, 30180, 53470, 93152, 159395, 268190, 444910, 727360, 1174563, 1873320, 2955010, 4611960, 7127305, 10912244, 16560430, 24924550, 37217620, 55160650, 81174270, 118651560, 172316445, 248718830, 356892660

Offset: 10

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001488, A022605, A327388, A338463, A341236, A341241, A341243, A341244, A341245, A341246, A341247, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, 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..44);  # Alois P. Heinz, Feb 07 2021

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^10.

cp's OEIS Frontend

A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

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A341251 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^9.

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A341241 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^3.

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A341244 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^5.

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A341246 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^7.

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A341247 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^8.

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A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

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A341265 Coefficient of x^(2*n) in (-1 + Product_{k>=1} 1 / (1 + x^k))^n.

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A341253 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^10.

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