A297329 -id:A297329 - cp's OEIS frontend

A297328 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^k.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 14, 0, 1, 5, 18, 37, 49, 25, 0, 1, 6, 25, 64, 114, 114, 56, 0, 1, 7, 33, 100, 219, 312, 282, 97, 0, 1, 8, 42, 146, 375, 676, 855, 624, 198, 0, 1, 9, 52, 203, 594, 1276, 2030, 2178, 1422, 354, 0, 1, 10, 63, 272, 889, 2196, 4155, 5736, 5496, 3058, 672, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 5)*x^2 + (1/6)*k*(k^2 + 15*k + 20)*x^3 + (1/24)*k*(k^3 + 30*k^2 + 155*k + 150)*x^4 + (1/120)*k*(k^4 + 50*k^3 + 575*k^2 + 1750*k + 624)*x^5 + ...
Square array begins:
  1,   1,    1,    1,    1,     1,  ...
  0,   1,    2,    3,    4,     5,  ...
  0,   3,    7,   12,   18,    25,  ...
  0,   6,   18,   37,   64,   100,  ...
  0,  14,   49,  114,  219,   375,  ...
  0,  25,  114,  312,  676,  1276,  ...

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

Columns k=0..32 give A000007, A006906, A022726, A022727, A022728, A022729, A022730, A022731, A022732, A022733, A022734, A022735, A022736, A022737, A022738, A022739, A022740, A022741, A022742, A022743, A022744, A022745, A022746, A022747, A022748, A022749, A022750, A022751, A022752, A022753, A022754, A022755, A022756.
Main diagonal gives A297329.
Antidiagonal sums give A299162.
Cf. A078308, A266941, A297321, A297323, A297325.

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

first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, 1/(1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017

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

A(0,k) = 1; A(n,k) = (k/n) * Sum_{j=1..n} A078308(j) * A(n-j,k). - Seiichi Manyama, Aug 16 2023

A297322 a(n) = [x^n] Product_{k>=1} (1 + k*x^k)^n.

Original entry on oeis.org

1, 1, 5, 28, 137, 726, 3896, 21071, 115089, 633007, 3500740, 19448573, 108458924, 606787572, 3404112479, 19142919543, 107874784017, 609021410570, 3443952349385, 19503777943838, 110599636109572, 627924447630011, 3568885868192419, 20304321490356084

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..1000

Main diagonal of A297321.

Cf. A297324, A297326, A297329.

f:= proc(n) local k;
coeff(series(mul((1+k*x^k)^n,k=1..n),x,n+1),x,n);
end proc:
map(f, [$0..30]); # Robert Israel, Dec 28 2017

Table[SeriesCoefficient[Product[(1 + k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]

a(n) = A297321(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 5.814548482877687529318372516965305077397562... and c = 0.2563102401728134539247148322678842806264... - Vaclav Kotesovec, Aug 01 2019

A297324 a(n) = [x^n] Product_{k>=1} (1 - k*x^k)^n.

Original entry on oeis.org

1, -1, -3, 8, 9, -26, -168, 489, 1041, -5599, 12, 27103, 23436, -222912, -435473, 3177433, 375569, -24956018, 6931209, 181844002, 57372644, -2158209675, 853739235, 20642183588, -25063980804, -148768035501, 224915906836, 1322267927471, -2337343745721, -12604818831294

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Main diagonal of A297323.

Cf. A297322, A297326, A297329.

Table[SeriesCoefficient[Product[(1 - k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 29}]

a(n) = A297323(n,n).

A297326 a(n) = [x^n] Product_{k>=1} 1/(1 + k*x^k)^n.

Original entry on oeis.org

1, -1, -1, -1, 27, -76, 95, -295, 2035, -8119, 22714, -66793, 254223, -988651, 3444055, -11402626, 39248691, -141740051, 511583207, -1798826901, 6256648862, -22054706773, 78889160635, -281698897727, 996551999479, -3520566280801, 12522382445455, -44731559517301

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

sign

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

Main diagonal of A297325.

Cf. A297322, A297324, A297329.

f:= proc(n) local k;
coeff(series(mul(1/(1+k*x^k)^n,k=1..n),x,n+1),x,n);
end proc:
map(f, [$0..30]); # Robert Israel, Dec 28 2017

Table[SeriesCoefficient[Product[1/(1 + k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

a(n) = A297325(n,n).

A298985 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.

Original entry on oeis.org

1, 1, 8, 54, 496, 5400, 73728, 1204322, 23167808, 512093178, 12781430600, 355128859129, 10863077554224, 362572265689777, 13107541496092960, 510105773344747725, 21258690342206888192, 944467894258279964254, 44555341678790400325512, 2224158766859058600584834, 117123916650423288611260400

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A000219, A124577, A261561, A261565, A266941, A297329, A298986, A298987, A298988.

b:= proc(n, i, k) option remember;   `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i+j-1, j)*b(n-i*j, i-1, k)*k^j, j=0..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 23 2018

Table[SeriesCoefficient[Product[1/(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ n^n. - Vaclav Kotesovec, Feb 02 2018

A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)^k).

Original entry on oeis.org

1, 1, 4, 16, 75, 366, 1887, 10010, 54493, 302302, 1703599, 9723774, 56101292, 326640411, 1916800425, 11325242328, 67316128903, 402245682741, 2414978550718, 14560379165160, 88122911824659, 535188028077586, 3260549998701951, 19921639754064470, 122041156818328779

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 16*x^3 + 75*x^4 + 366*x^5 + 1887*x^6 + 10010*x^7 + 54493*x^8 + 302302*x^9 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - 2*x^2*A(x)^2) * (1 - 3*x^3*A(x)^3) * ...).
log(A(x)) = x + 7*x^2/2 + 37*x^3/3 + 219*x^4/4 + 1276*x^5/5 + 7687*x^6/6 + 46551*x^7/7 + 285043*x^8/8 + ... + A297329(n)*x^n/n + ...

Cf. A006906, A109085, A297329, A301455, A301578.

cp's OEIS Frontend

A297328 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^k.

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A297322 a(n) = [x^n] Product_{k>=1} (1 + k*x^k)^n.

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A297324 a(n) = [x^n] Product_{k>=1} (1 - k*x^k)^n.

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A297326 a(n) = [x^n] Product_{k>=1} 1/(1 + k*x^k)^n.

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A298985 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.

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A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - k*x^k*A(x)^k).

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A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)^k).