A324595 -id:A324595 - cp's OEIS frontend

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

1, 1, 3, 13, 51, 206, 855, 3585, 15155, 64525, 276278, 1188353, 5130999, 22226049, 96544003, 420368858, 1834203955, 8018057345, 35107961175, 153950675585, 675978772326, 2971700764941, 13078268135683, 57613905606273, 254038914924791, 1121081799217231

Offset: 0

Vaclav Kotesovec, Mar 25 2016

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.

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:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),
       (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 31 2021

Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[(-1)^j*x^j/(j*(x^j - 1)), {j, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 26 2019

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...

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.

Original entry on oeis.org

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

A340987 Number of colored integer partitions of 2n such that all colors from an n-set are used.

Original entry on oeis.org

1, 2, 10, 59, 362, 2287, 14719, 95965, 631714, 4189334, 27946335, 187319827, 1260570515, 8511460908, 57634550179, 391232510284, 2661483301282, 18140003082945, 123846214549072, 846801764644618, 5797865791444367, 39745254613927264, 272762265331208465

Offset: 0

Alois P. Heinz, Feb 01 2021

nonn

			a(1) = 2: 2a, 1a1a.
a(2) = 10: 3a1b, 3b1a, 2a2b, 2a1b1b, 2b1a1a, 2a1a1b, 2b1a1b, 1a1b1b1b, 1a1a1b1b, 1a1a1a1b.

Alois P. Heinz, Table of n, a(n) for n = 0..1183

Cf. A000041, A060642, A144064, A324595.

b:= proc(n, k) option remember; `if`(k<2, combinat[numbpart](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..25);

b[n_, k_] := b[n, k] = If[k<2, PartitionsP[n+1], With[{q = Quotient[k, 2]}, Sum[b[j, q] b[n-j, k-q], {j, 0, n}]]];
a[n_] := b[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 04 2021, after Alois P. Heinz *)
Table[SeriesCoefficient[(-1 + 1/QPochhammer[Sqrt[x]])^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jan 15 2024 *)
(* Calculation of constant d: *) 1/r/.FindRoot[{1 + s == 1/QPochhammer[Sqrt[r*s]], 1/(1 + s) + Sqrt[r]*(1 + s)*Derivative[0, 1][QPochhammer][Sqrt[r*s], Sqrt[r*s]] / (2*Sqrt[s]) == (Log[1 - Sqrt[r*s]] + QPolyGamma[0, 1, Sqrt[r*s]]) / (s*Log[r*s])}, {r, 1/7}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 15 2024 *)

a(n) = [x^(2n)] (-1 + Product_{j>0} 1/(1-x^j))^n.
a(n) = A060642(2*n,n).
a(n) = Sum_{i=0..n} (-1)^i * C(n,i) * A144064(2n,n-i).
a(n) ~ c * d^n / sqrt(n), where d = 7.0224714601856191637116674203375767768930294104680988528373522936595686998... and c = 0.306577097117652483059452115503859901867921865482563952948772592499558... - Vaclav Kotesovec, Feb 14 2021

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

A341263 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 - x^k))^n.

Original entry on oeis.org

1, -1, 1, -1, -3, 19, -65, 181, -419, 755, -749, -1530, 12255, -47477, 141065, -343526, 660941, -770917, -911369, 9721976, -40135713, 124134772, -313463842, 631382751, -824406065, -492101356, 8192253811, -35948431288, 115087580857, -299576625051, 627027769120, -894734468883

Offset: 0

Ilya Gutkovskiy, Feb 07 2021

sign

Cf. A001482, A001483, A001484, A001485, A001486, A001487, A001488, A008705, A010815, A047654, A047655, A324595, A340987, A341265.

g:= proc(n) option remember; `if`(n=0, 1, add(add(
     -d, 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..31);  # Alois P. Heinz, Feb 07 2021

Table[SeriesCoefficient[(-1 + QPochhammer[x, x])^n, {x, 0, 2 n}], {n, 0, 31}]
A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] DivisorSigma[1, 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, 31}]

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

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

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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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A340987 Number of colored integer partitions of 2n such that all colors from an n-set are used.

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

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A341263 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 - x^k))^n.

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

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