A327380 -id:A327380 - cp's OEIS frontend

A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1

Offset: 0

Alois P. Heinz, Aug 29 2019

For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

T is the convolution triangle of A000009 (see A357368). - Peter Luschny, Oct 19 2022

			T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  3,  8,   9,   4,   1;
  0,  4, 14,  19,  14,   5,   1;
  0,  5, 22,  39,  36,  20,   6,   1;
  0,  6, 34,  72,  85,  60,  27,   7,  1;
  0,  8, 50, 128, 180, 160,  92,  35,  8, 1;
  0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.
Main diagonal and lower diagonals give: A000012, A001477, A000096.
Row sums give A304969.
T(2n,n) gives A324595.
Cf. A060642, A286335, A325915.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](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, b(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, Jan 31 2021
# Uses function PMatrix from A357368.
PMatrix(10, A000009); # Peter Luschny, Oct 19 2022

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,    b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).
Sum_{k=1..n} k * T(n,k) = A325915(n).
G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - Alois P. Heinz, Jan 29 2021

A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 5, 8, 9, 12, 15, 20, 23, 28, 36, 44, 52, 62, 76, 90, 106, 124, 149, 176, 203, 236, 279, 324, 372, 430, 499, 576, 657, 752, 867, 992, 1124, 1280, 1463, 1662, 1876, 2124, 2410, 2722, 3061, 3446, 3889, 4374, 4896, 5490, 6166, 6900, 7700, 8600

Offset: 2

Ilya Gutkovskiy, Jan 31 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1002

Cf. A000700, A022597, A047654, A048574, A073252, A327380, A338223.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (-1 + (&*[1+x^(2*j+1): j in [0..m+2]]) )^2 )); // G. C. Greubel, Sep 07 2023

nmax = 55; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
With[{k=2}, Drop[CoefficientList[Series[(2/QPochhammer[-1,-x] -1)^k, {x,0,80}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

m=80
def f(x): return (-1 + product(1+x^(2*j-1) for j in range(1,m+3)) )^2
def A338463_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
a=A338463_list(m); a[2:] # G. C. Greubel, Sep 07 2023

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^2.
a(n) = Sum_{k=1..n-1} A000700(k) * A000700(n-k).
a(n) = A073252(n) - 2 * A000700(n) for n > 0.
a(n) = [x^n]( (2/QPochhammer(-1,-x) - 1)^2 ). - G. C. Greubel, Sep 07 2023

A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 12, 30, 68, 144, 289, 556, 1034, 1868, 3292, 5678, 9608, 15984, 26188, 42314, 67509, 106460, 166090, 256552, 392628, 595696, 896484, 1338894, 1985298, 2923840, 4278448, 6222518, 8997544, 12938368, 18507297, 26340040, 37307326, 52597320, 73825504, 103180702

Offset: 2

Ilya Gutkovskiy, Jan 30 2021

nonn

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

Cf. A001934, A002448, A014968, A015128, A048574, A063725, A327380.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (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..37);  # Alois P. Heinz, Feb 10 2021

nmax = 37; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^2/4, {x, 0, nmax}], x] // Drop[#, 2] &
nmax = 37; CoefficientList[Series[(1/4) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
A015128[n_] := Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; a[n_] := (1/4) Sum[A015128[k] A015128[n - k], {k, 1, n - 1}]; Table[a[n], {n, 2, 37}]

G.f.: (1/4) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^2.
a(n) = Sum_{k=0..n} A014968(k) * A014968(n-k).
a(n) = (1/4) * Sum_{k=1..n-1} A015128(k) * A015128(n-k).
a(n) = (A001934(n) - 2 * A015128(n)) / 4 for n > 0.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.

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A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

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

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