A267007 -id:A267007 - cp's OEIS frontend

A162506 Convergent of an infinite product, abc,...; a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...],...

1, 1, 3, 6, 12, 23, 42, 77, 132, 236, 390, 664, 1087, 1782, 2858, 4601, 7216, 11344, 17650, 27162, 41632, 63316, 95717, 143558, 214644, 318464, 470879, 691968, 1012866, 1474434, 2140606, 3088874, 4445440, 6370142, 9095564, 12941289, 18350398, 25930984

Offset: 1

Gary W. Adamson, Jul 04 2009

nonn

Equals row sums of triangle A162507.

With offset 0, sum of products of parts, counted without multiplicity, in all partitions of n. Sum of products of parts, counted with multiplicity, in all partitions of n is A006906. - Vladeta Jovovic, Jul 24 2009

			First few rows of the array =
1,...1,...1,...1,...1,...
1,...1,...3,...3,...5,...
1,...1,...3,...6,...8,...
1,...1,...3,...6,..12,...
1,...1,...3,...6,..12,...
...tending to A162506: (1, 1, 3, 6, 12, 23, 42, 77, 132,...)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A162507, A267007.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-1)*i, j=1..n/i)))
    end:
a:= n-> b(n-1, n-1):
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 26 2013

nmax = 50; Rest[CoefficientList[Series[x*Product[1+k*x^k/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 08 2016 *)

Convergent of an infinite product, a*b*c,...; a = [1,1,1,...], b =
[1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...]; i.e. the infinite set of
sequences [1,...N,...,] interleaved with (N-2) adjacent zeros.
G.f.: x*Product(1+k*x^k/(1-x^k),k=1..infinity). - Vladeta Jovovic, Jul 24 2009

More terms from Vladeta Jovovic, Jul 22 2009

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]

A267008 Expansion of Product_{k>=1} (1 + (k+1)*x^k).

Original entry on oeis.org

1, 2, 3, 10, 13, 28, 58, 90, 146, 260, 481, 688, 1168, 1748, 2863, 4726, 6938, 10412, 16140, 23746, 35702, 55812, 79032, 116758, 168779, 247006, 350310, 513410, 744286, 1045466, 1485685, 2098780, 2935416, 4137878, 5746618, 8027612, 11343706, 15487222, 21418682

Offset: 0

Vaclav Kotesovec, Jan 08 2016

nonn

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

Cf. A022629, A074141, A267007.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 15 2019

nmax = 50; CoefficientList[Series[Product[1+(k+1)*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; Do[Do[poly[[j+1]] += (k+1)*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 13, 18, 37, 56, 101, 152, 285, 410, 713, 1118, 1830, 2780, 4618, 6934, 11278, 17092, 26894, 40822, 64435, 96372, 149299, 225104, 345131, 515394, 788176, 1169962, 1772957, 2632458, 3950365, 5849260, 8748993, 12867848, 19135894, 28126614, 41598695

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..8190
Vaclav Kotesovec, Closed form of the asymptotics

Cf. A006906, A034008, A052847, A074141, A267007.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
       0^n, b(n, i-1)+(i-1)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 19 2019

nmax = 40; CoefficientList[Series[Product[1/(1 - (k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[(j - 1)^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d - 1)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j - 1)^k*x^(j*k)/k).
From Vaclav Kotesovec, Sep 11 2018: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 28108804.248904780960402246466460350520790117596512766842168... if mod(n,5) = 0
c = 28108804.010850549080284030388905319123062152339902207992657... if mod(n,5) = 1
c = 28108804.067769166625741650205643600577757560110636366636106... if mod(n,5) = 2
c = 28108804.083581827971851596540314974909801290757084687583764... if mod(n,5) = 3
c = 28108804.058853893104368046896759214442695016905368229405793... if mod(n,5) = 4
(End)

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A162506 Convergent of an infinite product, a*b*c,...; a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...],...

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

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A267008 Expansion of Product_{k>=1} (1 + (k+1)*x^k).

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A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

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A162506 Convergent of an infinite product, abc,...; a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...],...