A022629 -id:A022629 - cp's OEIS frontend

A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 7, 3, 1, 1, 16, 35, 25, 15, 4, 1, 1, 32, 97, 91, 77, 25, 5, 1, 1, 64, 275, 337, 405, 161, 43, 6, 1, 1, 128, 793, 1267, 2177, 1069, 393, 64, 8, 1, 1, 256, 2315, 4825, 11925, 7313, 3799, 726, 120, 10

Offset: 0

Seiichi Manyama, Sep 11 2017

			Square array begins:
   1, 1,  1,  1,   1, ...
   1, 1,  1,  1,   1, ...
   1, 2,  4,  8,  16, ...
   2, 5, 13, 35,  97, ...
   2, 7, 25, 91, 337, ...

Alois P. Heinz, Rows n = 0..150, flattened

Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.
Rows 0+1, 2, 3 give A000012, A000079, A007689.
Main diagonal gives A292190.
Cf. A292166.

b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 11 2017

m = 14;
col[k_] := col[k] = Product[1 + j^k*x^j, {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.

Original entry on oeis.org

1, 1, 1, 5, 7, 37, 79, 173, 101, 127, 1033, 1571, 200069, 2564519, 5126711, 25661369, 532393, 431100529, 1855391, 1533985991, 48977868113, 342880481117, 342289639579, 435979161889, 1308720597671, 373092965489, 7824703695283, 24141028973, 31250466692609

Offset: 0

Alois P. Heinz, Dec 05 2018

a(n)/A322381(n) = A007838(n)/A000142(n) is the probability that a random permutation of [n] has distinct cycle sizes. - Geoffrey Critzer, Feb 23 2022

			1/1, 1/1, 1/2, 5/6, 7/12, 37/60, 79/120, 173/280, 101/168, 127/210, 1033/1680, 1571/2640, 200069/332640, 2564519/4324320, 5126711/8648640, ... = A322380/A322381

Alois P. Heinz, Table of n, a(n) for n = 0..1268
Andreas B. G. Blobel, An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k), arXiv:1904.07808 [math.CO], 2019.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 137.
A. Knopfmacher and J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.

Denominators: A322381.

Cf. A000009, A000142, A007838, A022629, A080130, A177208, A177209, A322364, A322365, A323290, A323291, A323339, A323340.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +b(n-i, min(i-1, n-i))/i))
    end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

Limit_{n->infinity} a(n)/A322381(n) = exp(-gamma) = A080130.

Sum_{n>=0} a(n)/A322381(n)*x^n = Product_{i>=1} (1 + x^i/i). - Geoffrey Critzer, Feb 23 2022

A266971 Expansion of Product_{k>=1} 1 / (1 + k*x^k)^k.

Original entry on oeis.org

1, -1, -3, -6, 2, 9, 41, 46, 91, -110, -210, -713, -574, -1152, 792, 1066, 9317, 8553, 21302, 745, 8051, -82940, -76750, -276022, -82369, -404100, 381095, -38110, 2427272, 1126260, 6527840, 198507, 9754305, -14320206, 2879362, -60271740, -5154261, -143468194

Offset: 0

Vaclav Kotesovec, Jan 07 2016

sign

For n > 36 is a(n) > 0 if n is even and a(n) < 0 if n is odd.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = -n. - Seiichi Manyama, Nov 18 2017

Seiichi Manyama, Table of n, a(n) for n = 0..6224 (terms 0..1000 from Vaclav Kotesovec)

Cf. A022629, A022693, A266891, A266941, A266964.

nmax=50; CoefficientList[Series[Product[1/(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+k*x^k)^k)) \\ Seiichi Manyama, Nov 18 2017

def s(f_ary, g_ary, n)
  s = 0
  (1..n).each{|i| s += i * f_ary[i] * g_ary[i] ** (n / i) if n % i == 0}
  s
end
def A(f_ary, g_ary, n)
  ary = [1]
  a = [0] + (1..n).map{|i| s(f_ary, g_ary, i)}
  (1..n).each{|i| ary << (1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
  ary
end
def A266971(n)
  A((0..n).to_a, (0..n).map{|i| -i}, n)
end
p A266971(50) # Seiichi Manyama, Nov 18 2017

a(n) ~ c * (-1)^n * n^2 * 3^(n/3), where
c = 50.5838262902886367070621... if mod(n,3)=0,
c = 50.5827771239052189170531... if mod(n,3)=1,
c = 50.5832885870455104598393... if mod(n,3)=2.
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(-d)^(n/d). - Seiichi Manyama, Nov 18 2017

A267004 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k)).

Original entry on oeis.org

1, 2, 5, 12, 24, 50, 97, 184, 331, 606, 1061, 1834, 3125, 5228, 8673, 14250, 23034, 36894, 58750, 92298, 144398, 223994, 344916, 527116, 801295, 1209870, 1816539, 2713956, 4033169, 5961700, 8775236, 12852444, 18742153, 27225316, 39371647, 56743200, 81467211

Offset: 0

Vaclav Kotesovec, Jan 08 2016

nonn

Convolution of A022629 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A022629, A267005, A267007.

nmax = 50; CoefficientList[Series[Product[(1+k*x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 2, 1, 1, 1, 5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 5, 5, 4, 3, 3, 3, 2, 2, 1, 1, 1, 7, 6, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 8, 7, 6, 6, 4, 4, 4, 3, 2, 2, 1, 1

Offset: 1

Clark Kimberling

Row sums yield A015723. T(n,1)=A025147(n-1); T(n,2)=A015744(n-2); T(n,3)=A015745(n-3); T(n,4)=A015746(n-4); T(n,5)=A015750(n-5). - Emeric Deutsch, Mar 29 2006

Number of parts of size k in all partitions of n into distinct parts. Number of partitions of n-k into distinct parts not including a part of size k. - Franklin T. Adams-Watters, Jan 24 2012

			T(8,3)=2 because we have [5,3] and [4,3,1].
Triangle begins:
n/k 1 2 3 4 5 6 7 8 9 10
01: 1
02: 0 1
03: 1 1 1
04: 1 0 1 1
05: 1 1 1 1 1
06: 2 2 1 1 1 1
07: 2 2 1 2 1 1 1
08: 3 2 2 1 2 1 1 1
09: 3 3 3 2 2 2 1 1 1
10: 5 4 4 3 2 2 2 1 1 1
...
The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), which is row n = 6. - _Gus Wiseman_, May 07 2019

Mircea Merca, Table of n, a(n) for n = 1..7260

Cf. A015723, A015744, A015745, A015746, A015750, A025147, A027293, A066633.

Cf. A006128, A022629, A066189, A246867, A325504, A325506, A325513.

g:=product(1+x^j,j=1..50)*sum(t^i*x^i/(1+x^i),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 29 2006
seq(seq(coeff(x^k*(product(1+x^j, j=1..n))/(1+x^k), x, n), k=1..n), n=1..13); # Mircea Merca, Feb 28 2014

z = 15; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; p[n_, k_] := p[n, k] = d[n][[k]]; s[n_] := s[n] = Flatten[Table[p[n, k], {k, 1, PartitionsQ[n]}]]; t[n_, k_] := Count[s[n], k]; u = Table[t[n, k], {n, 1, z}, {k, 1, n}]; TableForm[u] (* A015716 as a triangle *)
v = Flatten[u] (* A015716 as a sequence *)
(* Clark Kimberling, Mar 14 2014 *)

G.f.: G(t,x) = Product_{j>=1} (1+x^j) * Sum_{i>=1} t^i*x^i/(1+x^i). - Emeric Deutsch, Mar 29 2006
From Mircea Merca, Feb 28 2014: (Start)
a(n) = A238450(n) + A238451(n).
T(n,k) = Sum_{j=1..floor(n/k)} (-1)^(j-1)*A000009(n-j*k).
G.f.: for column k: q^k/(1+q^k)*(-q;q)_{inf}. (End)

A265840 Expansion of Product_{k>=1} (1 + k^3*x^k).

Original entry on oeis.org

1, 1, 8, 35, 91, 405, 1069, 3799, 8686, 36744, 86310, 235776, 686329, 1605779, 5230579, 13191702, 30608501, 73907925, 206052723, 433747560, 1324608945, 2995740974, 6973434054, 15364943439, 35816669079, 86662644756, 184871083828, 502089539734, 1098571699830

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A022629, A092484, A265837, A265841, A265842.

Column k=3 of A292189.

nmax = 40; CoefficientList[Series[Product[1 + k^3*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ 3*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A265841 Expansion of Product_{k>=1} (1 + k^4*x^k).

Original entry on oeis.org

1, 1, 16, 97, 337, 2177, 7313, 38529, 108594, 717186, 2053522, 7527458, 30757155, 88042387, 448973459, 1390503396, 4087546309, 12699966117, 49599776261, 124699632310, 608410782855, 1651128186296, 4862631132392, 13170300313769, 39285370060347, 130999461143020

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A022629, A092484, A265838, A265840, A265842.

Column k=4 of A292189.

nmax = 40; CoefficientList[Series[Product[1 + k^4*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(4*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ 4*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A265842 Expansion of Product_{k>=1} (1 + k^5*x^k).

Original entry on oeis.org

1, 1, 32, 275, 1267, 11925, 51445, 406183, 1406614, 14690040, 51144366, 251885088, 1481359033, 5108404955, 42614629915, 158222158038, 588574803125, 2360755022421, 13255325882835, 39266011999104, 325719196861377, 1031732678138822, 3791401325667894

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A022629, A092484, A265839, A265840, A265841.

Column k=5 of A292189.

nmax = 40; CoefficientList[Series[Product[1 + k^5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(5*k)*x^(j*k)/k). - Ilya Gutkovskiy, Oct 18 2018

Conjecture: log(a(n)) ~ 5*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A268500 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k)).

Original entry on oeis.org

1, 0, 1, 2, 2, 6, 7, 14, 11, 42, 39, 70, 95, 142, 239, 378, 418, 624, 1106, 1200, 2250, 2836, 4166, 4902, 8021, 10410, 14961, 21268, 29477, 36714, 54172, 68358, 95071, 134946, 168035, 254190, 322335, 427338, 541054, 787264, 964969, 1340730, 1748094, 2311386

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A022629, A267004, A267005, A268498, A268499.

nmax = 50; CoefficientList[Series[Product[(1+k*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

A318416 Expansion of Product_{i>=1, j>=1} (1 + ijx^(i*j)).

Original entry on oeis.org

1, 1, 4, 10, 22, 50, 115, 231, 470, 995, 1912, 3745, 7222, 13608, 25345, 47322, 85654, 155163, 278867, 494080, 870618, 1524769, 2640527, 4549564, 7802037, 13251684, 22412317, 37706268, 63015263, 104800015, 173574936, 285694401, 468449681, 764775169, 1242535747, 2010866469, 3242127656

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A022629, A107742, A280541, A318415.

a:=series(mul(mul(1+i*j*x^(i*j),j=1..55),i=1..55),x=0,37): seq(coeff(a,x,n),n=0..36); # Paolo P. Lava, Apr 02 2019

nmax = 36; CoefficientList[Series[Product[Product[(1 + i j x^(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Product[(1 + k x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Exp[Sum[Sum[(-d)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-d)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 36}]
nmax = 36; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 27 2018 *)

G.f.: Product_{k>=1} (1 + k*x^k)^tau(k), where tau = number of divisors (A000005).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-d)^(k/d+1)*tau(d) ) * x^k/k).

cp's OEIS Frontend

A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A266971 Expansion of Product_{k>=1} 1 / (1 + k*x^k)^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Ruby

Formula

A267004 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A265840 Expansion of Product_{k>=1} (1 + k^3*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A265841 Expansion of Product_{k>=1} (1 + k^4*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A265842 Expansion of Product_{k>=1} (1 + k^5*x^k).

Views

Author

Keywords

Links

Crossrefs

A318416 Expansion of Product_{i>=1, j>=1} (1 + ijx^(i*j)).