A014153 -id:A014153 - cp's OEIS frontend

A103628 Total sum of parts of multiplicity 1 in all partitions of n.

0, 1, 2, 6, 10, 21, 33, 59, 89, 145, 212, 325, 463, 680, 948, 1348, 1845, 2558, 3446, 4681, 6219, 8306, 10901, 14352, 18632, 24230, 31151, 40077, 51074, 65088, 82290, 103986, 130517, 163679, 204078, 254174, 314975, 389839, 480369, 591133, 724600, 886965

Offset: 0

Vladeta Jovovic, Mar 25 2005

Total number of parts of multiplicity 1 in all partitions of n is A024786(n+1).

Equals A000041 convolved with A026741. - Gary W. Adamson, Jun 11 2009

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and a(4) = 0 + 2 + 0 + (1+3) + 4 = 10.

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

Cf. A026741. - Gary W. Adamson, Jun 11 2009

Column k=1 of A222730. - Alois P. Heinz, Mar 03 2013

gf:=x*(1+x+x^2)/(1-x^2)^2/product((1-x^k), k=1..500): s:=series(gf, x, 100): for n from 0 to 60 do printf(`%d,`,coeff(s, x, n)) od: # James Sellers, Apr 22 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], add((l->`if`(j=1, [l[1],
       l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2013

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[3]]; a[0] = 0; Table[a[n], {n, 0, 50}]  (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: x*(1+x+x^2)/(1-x^2)^2 /Product_{k>0}(1-x^k).
a(n) = A066186(n) - A194544(n). - Omar E. Pol, Nov 20 2011
a(n) = 3*A014153(n)/4 - 3*A000070(n)/4 - A270143(n+1)/4 + A087787(n)/4. - Vaclav Kotesovec, Nov 05 2016
a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2) * (1 - Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Nov 05 2016

More terms from James Sellers, Apr 22 2005

A210952 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the partitions of n but with the partitions aligned to the right margin.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 7, 9, 1, 3, 7, 12, 12, 1, 3, 7, 14, 21, 20, 1, 3, 7, 14, 24, 31, 25, 1, 3, 7, 14, 26, 40, 47, 38, 1, 3, 7, 14, 26, 43, 61, 66, 49, 1, 3, 7, 14, 26, 45, 70, 92, 93, 69, 1, 3, 7, 14, 26, 45, 73, 106, 130, 124, 87, 1, 3, 7, 14

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 the illustration shows the partitions of 6 aligned to the right margin and below the sums of the columns:
.
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                  5 + 1
.              3 + 2 + 1
.              4 + 1 + 1
.          2 + 2 + 1 + 1
.          3 + 1 + 1 + 1
.      2 + 1 + 1 + 1 + 1
.  1 + 1 + 1 + 1 + 1 + 1
-------------------------
.  1,  3,  7, 14, 21, 20
.
So row 6 lists 1, 3, 7, 14, 21, 20.
Triangle begins:
1;
1, 3;
1, 3, 5;
1, 3, 7,  9;
1, 3, 7, 12, 12;
1, 3, 7, 14, 21, 20;
1, 3, 7, 14, 24, 31, 25;
1, 3, 7, 14, 26, 40, 47, 38;
1, 3, 7, 14, 26, 43, 61, 66, 49;
1, 3, 7, 14, 26, 45, 70, 92, 93, 69:

Mirror of triangle A206283. Rows sums give A066186. Rows converge to A014153. Right border gives A046746, >= 1.

Cf. A135010, A138121, A181187, A210763, A210950, A210951, A210953, A210961, A210970.

T(n,k) = Sum_{j=1..n} A210953(j,k). - Omar E. Pol, May 26 2012

A341062 Sequence whose partial sums give A000005.

Original entry on oeis.org

1, 1, 0, 1, -1, 2, -2, 2, -1, 1, -2, 4, -4, 2, 0, 1, -3, 4, -4, 4, -2, 0, -2, 6, -5, 1, 0, 2, -4, 6, -6, 4, -2, 0, 0, 5, -7, 2, 0, 4, -6, 6, -6, 4, 0, -2, -2, 8, -7, 3, -2, 2, -4, 6, -4, 4, -4, 0, -2, 10, -10, 2, 2, 1, -3, 4, -6, 4, -2, 4, -6, 10, -10, 2, 2, 0, -2, 4, -6, 8, -5, -1, -2, 10, -8, 0, 0, 4, -6, 10

Offset: 1

Omar E. Pol, Feb 04 2021

Essentially a duplicate of A051950.
Convolved with A000041 gives A138137.
Convolved with A000027 gives the nonzero terms of A006218.
Convolved with A000070 gives the nonzero terms of A006128.
Convolved with A014153 gives the nonzero terms of A284870.
Convolved with A036469 gives the nonzero terms of A305082.
Convolved with the nonzero terms of A006218 gives A055507.
Convolved with the nonzero terms of A000217 gives the nonzero terms of A078567.

1 together with A051950.
Cf. A000005 (partial sums).
Cf. A000027, A000041, A000070, A000217, A006128, A006218, A014153, A036469, A055507, A078567, A138137, A284870, A305082, A340793.

Join[{1}, Differences[Table[DivisorSigma[0, n], {n, 1, 90}]]] (* Amiram Eldar, Feb 06 2021 *)

a(n) = A051950(n) for n > 1.

A000412 Number of bipartite partitions of n white objects and 3 black ones.

Original entry on oeis.org

3, 7, 16, 31, 57, 97, 162, 257, 401, 608, 907, 1325, 1914, 2719, 3824, 5313, 7316, 9973, 13495, 18105, 24132, 31938, 42021, 54948, 71484, 92492, 119120, 152686, 194887, 247693, 313613, 395547, 497154, 622688, 777424, 967525, 1200572, 1485393, 1832779, 2255317

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n*q^3 where p and q are distinct primes.
Number of Gaussian partitions of n+3*i or 3+n*i where a "Gaussian partition" is a way of writing a Gaussian integer with nonnegative parts as a sum of Gaussian integers with nonnegative parts, imaginary numbers and real numbers. For k = 3+1*i (where i is the imaginary unit), the a(1)=7 ways to write k (where parentheses represent a complex number and a lack of them represents a sum of a real and imaginary number) would be 3+i, (3+i), 2+1+i, (2+i)+1, (1+i)+2, 1+1+1+i, (1+i)+1+1. - Yali Harrary, Nov 20 2022
a(n) is the number of multiset partitions of the multiset {r^n, s^3}. - Joerg Arndt, Jan 01 2024

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..100 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953), pp. 72-83.
F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956 (annotated scanned pages from, plus a review).

Column 3 of A054225.

Cf. A005380.

max = 40; col = 3; s1 = Series[Product[1/(1-x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}] // Normal; s2 = Series[s1, {x, 0, max+1}]; a[n_] := SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[ a[n] , {n, 0, max}] (* Jean-François Alcover, Mar 13 2014 *)
nmax = 50; CoefficientList[Series[(3 + x - x^2 - 2*x^3 - x^4 + x^5)/((1-x)*(1-x^2)*(1-x^3)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

a(n) = if n <= 3 then A054225(3,n), otherwise a(n) = A054225(n,3). - Reinhard Zumkeller, Nov 30 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * sqrt(n) / (2*sqrt(2)*Pi^3). - Vaclav Kotesovec, Feb 01 2016
a(n) = A000098(n) + A000070(n) + A014153(n). - Yali Harrary, Nov 20 2022

Edited by Christian G. Bower, Jan 08 2004

A014160 Apply partial sum operator thrice to partition numbers.

Original entry on oeis.org

1, 4, 11, 25, 51, 96, 171, 291, 478, 762, 1185, 1803, 2693, 3956, 5727, 8182, 11552, 16134, 22313, 30579, 41559, 56045, 75039, 99796, 131891, 173282, 226405, 294270, 380595, 489945, 627924, 801374, 1018644

Offset: 0

N. J. A. Sloane

nonn

A014160 convolved with A010815 = A000217, the triangular numbers. - Gary W. Adamson, Nov 09 2008
Unordered partitions of n into parts where the part 1 comes in 4 colors. - Peter Bala, Dec 23 2013
From Omar E. Pol, Mar 01 2023: (Start)
Partial sums of A014153.
Convolution of A000070 and A000027.
Convolution of A000041 and the positive terms of A000217.
Convolution of A002865 and the positive terms of A000292. (End)

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

Cf. A000041, A000070, A014153.
Cf. A010815, A000217. - Gary W. Adamson, Nov 09 2008
Column k=4 of A292508.
Cf. A000027, A002865, A000292.

nmax = 50; CoefficientList[Series[1/((1-x)^3 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

From Peter Bala, Dec 23 2013: (Start)
O.g.f.: 1/(1 - x)^3 * Product_{k >= 1} 1/(1 - x^k).
a(n-1) + a(n-2) = Sum_{parts k in all partitions of n} J_2(k), where J_2(n) is the Jordan totient function A007434(n). (End)
a(n) ~ 3*sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2)*Pi^3). - Vaclav Kotesovec, Oct 30 2015
a(n) = Sum_{k=0..n} A014153(k). - Sean A. Irvine, Oct 14 2018

A284870 Expansion of Sum_{i>=1} ix^i/(1 - x) Product_{j=1..i} 1/(1 - x^j).

Original entry on oeis.org

0, 1, 4, 10, 22, 42, 77, 131, 217, 345, 537, 812, 1211, 1767, 2547, 3615, 5078, 7043, 9687, 13185, 17815, 23867, 31766, 41972, 55146, 71997, 93519, 120813, 155358, 198811, 253374, 321509, 406436, 511802, 642264, 803140, 1001154, 1243966, 1541167, 1903754, 2345300, 2881404, 3531195, 4316632, 5264444, 6405389

Offset: 0

Ilya Gutkovskiy, Apr 04 2017

nonn

Total number of parts in all partitions of all positive integers <= n.
Sum of largest parts of all partitions of all positive integers <= n.
From Omar E. Pol, Feb 16 2021: (Start)
Apart from initial zero this is as follows:
Convolution of A341062 and A014153.
Convolution of A000005 and A000070.
Convolution of nonzero terms of A006218 and A000041.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A340581. These divisors are also all parts of all partitions of all positive integers <= n. (End)

			a(4) = 22 because we have 1 = 1, 2 = 2, 1 + 1 = 2, 3 = 3, 2 + 1 = 3, 1 + 1 + 1 = 3, 4 = 4, 3 + 1 = 4, 2 + 2 = 4, 2 + 1 + 1 = 4 and 1 + 1 + 1 + 1 = 4 therefore 1 + 1 + 2 + 1 + 2 + 3 + 1 + 2 + 2 + 3 + 4 = 22 (total number of parts) or 1 + 2 + 1 + 3 + 2 + 1 + 4 + 3 + 2 + 2 + 1 = 22 (sum of largest parts).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000070, A006128, A015724, A026905, A124920, A138137, A182738, A299779.

Cf. A000041, A006218, A014153, A340581, A341062.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1) +(p-> p+[0, p[1]])(b(n-i, min(n-i, i))))
    end:
a:= proc(n) a(n):= `if`(n<1, 0, a(n-1)+b(n$2)[2]) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Feb 16 2021

nmax = 45; CoefficientList[Series[Sum[i x^i /(1 - x) Product[1/(1 - x^j), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[1/(1 - x) Sum[x^i /(1 - x^i), {i, 1, nmax}] Product[1/(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Accumulate[Table[Sum[DivisorSigma[0, k] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]]

G.f.: Sum_{i>=1} i*x^i/(1 - x) * Product_{j=1..i} 1/(1 - x^j).
G.f.: (1/(1 - x)) * Sum_{i>=1} x^i/(1 - x^i) * Product_{j>=1} 1/(1 - x^j).
a(n) = Sum_{k=0..n} A006128(k).
a(n) = A124920(n+1) - 1.
a(n) = Sum_{k=1..n} k * A299779(n,k). - Alois P. Heinz, May 14 2018

A055375 Euler transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 14, 21, 14, 5, 7, 26, 48, 48, 26, 7, 11, 45, 103, 131, 103, 45, 11, 15, 75, 198, 312, 312, 198, 75, 15, 22, 120, 366, 674, 830, 674, 366, 120, 22, 30, 187, 637, 1359, 1961, 1961, 1359, 637, 187, 30, 42, 284, 1078, 2584, 4302, 5066, 4302, 2584, 1078, 284, 42

Offset: 0

Christian G. Bower, May 16 2000

Number of partitions of n objects, k of which are black, into parts each of which is a sequence of objects. E.g. T(3,1) = 7; the partitions are [BWW], [WBW], [WWB], [BW,W], [WB,W], [WW,B] and [B,W,W]. - Franklin T. Adams-Watters, Jan 10 2007

			Triangle begins
   1;
   1,  1;
   2,  3,   2;
   3,  7,   7,   3;
   5, 14,  21,  14,   5;
   7, 26,  48,  48,  26,   7;
  11, 45, 103, 131, 103,  45, 11;
  15, 75, 198, 312, 312, 198, 75, 15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A034899.
Columns k=0-1 give: A000041, A014153(n-1) for n>=1.
T(2n,n) gives A360626.
Cf. A007318, A034691, A209406, A360634.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 14 2023

nmax = 10; pp = Product[Product[1/(1 - x^i*y^j)^Binomial[i, j], {j, 0, i}], {i, 1, nmax}]; t[n_, k_] := SeriesCoefficient[pp, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2017 *)

G.f.: Product_{i>=1} Product_{j=0..i} 1/(1 - x^i y^j)^C(i,j). - Franklin T. Adams-Watters, Jan 10 2007

Sum_{k=0..2n} (-1)^k * T(2n,k) = A034691(n). - Alois P. Heinz, Dec 05 2023

A206283 Triangle read by rows: T(n,k) = sum of the k-th parts of all partitions of n with their parts written in nondecreasing order.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 9, 7, 3, 1, 12, 12, 7, 3, 1, 20, 21, 14, 7, 3, 1, 25, 31, 24, 14, 7, 3, 1, 38, 47, 40, 26, 14, 7, 3, 1, 49, 66, 61, 43, 26, 14, 7, 3, 1, 69, 93, 92, 70, 45, 26, 14, 7, 3, 1, 87, 124, 130, 106, 73, 45, 26, 14, 7, 3, 1

Offset: 1

Omar E. Pol, Feb 13 2012

In row n, the sum of all odd-indexed terms minus the sum of all even-indexed terms is equal to A194714(n).

Reversed rows converge to A014153. - Alois P. Heinz, Feb 13 2012

			Row 4 is 9, 7, 3, 1 because the five partitions of 4, with their parts written in nondecreasing order, are
.                               4
.                               1, 3
.                               2, 2
.                               1, 1, 2
.                               1, 1, 1, 1
-------------------------------------------
And the sums of the columns are 9, 7, 3, 1.
.
Triangle begins:
   1;
   3,  1;
   5,  3,  1;
   9,  7,  3,  1;
  12, 12,  7,  3,  1;
  20, 21, 14,  7,  3,  1;
  25, 31, 24, 14,  7,  3,  1;
  38, 47, 40, 26, 14,  7,  3,  1;
  49, 66, 61, 43, 26, 14,  7,  3,  1;
  69, 93, 92, 70, 45, 26, 14,  7,  3,  1;

Alois P. Heinz, Rows n = 1..70, flattened

Column 1 is A046746. Row sums give A066186.

Cf. A014153, A181187, A194714.

More terms from Alois P. Heinz, Feb 13 2012

A210953 Triangle read by rows: T(n,k) = sum of all parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

Original entry on oeis.org

1, 0, 3, 0, 0, 5, 0, 0, 2, 9, 0, 0, 0, 3, 12, 0, 0, 0, 2, 9, 20, 0, 0, 0, 0, 3, 11, 25, 0, 0, 0, 0, 2, 9, 22, 38, 0, 0, 0, 0, 0, 3, 14, 28, 49, 0, 0, 0, 0, 0, 2, 9, 26, 44, 69, 0, 0, 0, 0, 0, 0, 3, 14, 37, 55, 87, 0, 0, 0, 0, 0, 0, 2, 9, 29, 62, 83, 123

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 and k = 1..6 the 6th shell looks like this:
-------------------------
k: 1,  2,  3,  4,  5,  6
-------------------------
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.
The sums of all parts in columns 1-6 are
.  0,  0,  0,  2,  9, 20, the same as the 6th row of triangle.
Triangle begins:
1;
0, 3;
0, 0, 5;
0, 0, 2, 9;
0, 0, 0, 3, 12;
0, 0, 0, 2,  9, 20;
0, 0, 0, 0,  3, 11, 25;
0, 0, 0, 0,  2,  9, 22, 38;
0, 0, 0, 0,  0,  3, 14, 28, 49;
0, 0, 0, 0,  0,  2,  9, 26, 44, 69;
0, 0, 0, 0,  0,  0,  3, 14, 37, 55, 87;
0, 0, 0, 0,  0,  0,  2,  9, 29, 62, 83, 123;

Row sums give A138879. Column sums converge to A014153. Right border gives A046746, n >= 1.

Cf. A135010, A138121, A210950, A210951, A210952.

A336516 Sum of parts, counted without multiplicity, in all compositions of n.

Original entry on oeis.org

0, 1, 3, 10, 24, 59, 136, 309, 682, 1493, 3223, 6904, 14675, 31013, 65202, 136512, 284748, 592082, 1227709, 2539516, 5241640, 10798133, 22206568, 45597489, 93495667, 191464970, 391636718, 800233551, 1633530732, 3331568080, 6789078236, 13824212219, 28129459098

Offset: 0

Alois P. Heinz, Jul 24 2020

nonn

			a(4) = 1 + 1 + 2 + 1 + 2 + 1 + 2 + 2 + 1 + 3 + 3 + 1 + 4 = 24: (1)111, (1)1(2), (1)(2)1, (2)(1)1, (2)2, (1)(3), (3)(1), (4).

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

Cf. A001787 (all parts), A014153 (the same for partitions), A336511, A336512, A336579, A336875.

b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
      `if`(i<1, 0, add((p-> [0, `if`(j=0, 0, p[1]*i)]+p)(
         b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
     If[i < 1, {0, 0}, Sum[Function[{0, If[j == 0, 0, #[[1]]*i]} + #][
       b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A103628 Total sum of parts of multiplicity 1 in all partitions of n.

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A210952 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the partitions of n but with the partitions aligned to the right margin.

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A341062 Sequence whose partial sums give A000005.

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A000412 Number of bipartite partitions of n white objects and 3 black ones.

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A014160 Apply partial sum operator thrice to partition numbers.

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A284870 Expansion of Sum_{i>=1} i*x^i/(1 - x) * Product_{j=1..i} 1/(1 - x^j).

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A055375 Euler transform of Pascal's triangle A007318.

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A206283 Triangle read by rows: T(n,k) = sum of the k-th parts of all partitions of n with their parts written in nondecreasing order.

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A284870 Expansion of Sum_{i>=1} ix^i/(1 - x) Product_{j=1..i} 1/(1 - x^j).