A293554 -id:A293554 - cp's OEIS frontend

A305206 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k/(k(1 - x^k)^n)).

1, 1, 2, 9, 36, 190, 1070, 6797, 46942, 350901, 2806187, 23894662, 215598410, 2053090936, 20557071012, 215697357449, 2364810631734, 27023086395647, 321160376470277, 3962047673946906, 50648323260067319, 669819485900273336, 9150740338219903590, 128965789655207156299

Offset: 0

Ilya Gutkovskiy, May 27 2018

nonn

N. J. A. Sloane, Transforms

Cf. A026007, A028377, A258343, A293554, A305205.

Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(n+k-2,n-1).

A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 141, 86, 22, 1, 1, 1, 9, 36, 105, 216, 331, 310, 160, 30, 1, 1, 1, 10, 45, 148, 357, 672, 855, 692, 282, 42, 1, 1, 1, 11, 55, 201, 554, 1232, 1982, 2214, 1483, 500, 56, 1

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

A(n,k) is the Euler transform of j -> binomial(j+k-2,k-1) evaluated at n.

			Square array begins:
1,  1,   1,   1,    1,    1,  ...
1,  1,   1,   1,    1,    1,  ...
1,  2,   3,   4,    5,    6,  ...
1,  3,   6,  10,   15,   21,  ...
1,  5,  13,  26,   45,   71,  ...
1,  7,  24,  59,  120,  216,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
S. Balakrishnan, S. Govindarajan, and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
N. J. A. Sloane, Transforms

Columns k=0..8 give A000012, A000041, A000219, A000294, A000335, A000391, A000417, A000428, A255965.
Main diagonal gives A293554.
Cf. A007318, A096751 (a similar but different sequence).

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-2, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 17 2017

Table[Function[k, SeriesCoefficient[E^(Sum[x^i/(i (1 - x^i)^k), {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

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

For asymptotics of column k see comment from Vaclav Kotesovec in A255965.

A305205 a(n) = [x^n] exp(-Sum_{k>=1} x^k/(k*(1 - x^k)^n)).

Original entry on oeis.org

1, -1, -2, -3, -4, 30, 274, 1841, 9358, 32463, -41557, -2265846, -28939286, -272101778, -2038274408, -10494221259, 9056975574, 1244820826687, 22703501504125, 299864024917632, 3221417281127823, 26849622543478562, 110101743392268978, -1810492304600468063

Offset: 0

Ilya Gutkovskiy, May 27 2018

sign

N. J. A. Sloane, Transforms

Cf. A073592, A292386, A292387, A293554, A305206.

Table[SeriesCoefficient[Exp[-Sum[x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 - x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} (1 - x^k)^binomial(n+k-2,n-1).

A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 4, 14, 65, 323, 1890, 12002, 83901, 630818, 5081318, 43546333, 395422430, 3788368227, 38151667046, 402516707510, 4436230390977, 50948789415297, 608433141666219, 7540823673023319, 96826154085714992, 1285991546051286085, 17640769457638701839, 249602608552024560609

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000990, A023871, A253289, A279215, A293554, A305206, A305653, A305655.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

A305655 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + x^k)/(k*(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 3, 13, 54, 290, 1674, 10857, 76398, 580230, 4706734, 40598349, 370694845, 3569027696, 36100349833, 382360758863, 4228730647420, 48716663849192, 583403253712747, 7248883337962522, 93291181556742684, 1241632098163126324, 17064777292709034968, 241874821482784132204

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

N. J. A. Sloane, Transforms

Cf. A027998, A255835, A281156, A293554, A305206, A305654.

Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 + x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} (1 + x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

A305255 a(n) = [x^n] exp(Sum_{k>=1} (-1)^kx^k/(k(1 - x^k)^n)).

Original entry on oeis.org

1, -1, -1, -4, -3, 14, 240, 1686, 9479, 36761, 3412, -1951731, -27296124, -268495319, -2093667873, -11586874946, -3788945531, 1127535019748, 21900095232973, 297591401221473, 3270627818325128, 28116733997044842, 129815302615081267, -1568168714539146596, -59839621829784309343

Offset: 0

Ilya Gutkovskiy, May 28 2018

sign

N. J. A. Sloane, Transforms

Cf. A255528, A293554, A294846, A305205, A305206.

Table[SeriesCoefficient[Exp[Sum[(-1)^k x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[1/(1 + x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 24}]

a(n) = [x^n] Product_{k>=1} 1/(1 + x^k)^binomial(n+k-2,n-1).

cp's OEIS Frontend

A305206 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^n)).

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A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

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A305205 a(n) = [x^n] exp(-Sum_{k>=1} x^k/(k*(1 - x^k)^n)).

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A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^n)).

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Mathematica

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A305655 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^n)).

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Programs

Mathematica

Formula

A305255 a(n) = [x^n] exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^n)).

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Author

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Programs

Mathematica

Formula

A305206 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k/(k(1 - x^k)^n)).

A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).

A305655 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + x^k)/(k*(1 - x^k)^n)).

A305255 a(n) = [x^n] exp(Sum_{k>=1} (-1)^kx^k/(k(1 - x^k)^n)).