A014153 -id:A014153 - cp's OEIS frontend

A120477 Apply partial sum operator 5 times to partition numbers.

1, 6, 22, 63, 155, 343, 702, 1352, 2480, 4370, 7445, 12323, 19894, 31421, 48675, 74111, 111099, 164221, 239656, 345670, 493243, 696861, 975518, 1353971, 1864315, 2547941, 3457972, 4662273, 6247169, 8322010, 11024775, 14528914, 19051697

Offset: 0

Jonathan Vos Post, Jul 21 2006

nonn

In general, if g.f. = 1/(1-x)^m * Product_{k>=1} 1/(1-x^k), then a(n) ~ 2^(m/2 - 2) * 3^((m-1)/2) * n^(m/2 - 1) * exp(Pi*sqrt(2*n/3)) / Pi^m. - Vaclav Kotesovec, Oct 30 2015

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

Cf. A000041, A000070, A014153, A014160, A014161.

Column k=6 of A292508.

with(combinat): g:=1/(1-x)^5/product(1-x^k,k=1..50): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..37); # Emeric Deutsch, Jul 24 2006

nmax = 50; CoefficientList[Series[1/((1-x)^5 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

G.f.: 1/((1-x)^5*Product_{k>=1} (1-x^k)). - Emeric Deutsch, Jul 24 2006

a(n) ~ 9*sqrt(2)*n^(3/2) * exp(Pi*sqrt(2*n/3)) / Pi^5. - Vaclav Kotesovec, Oct 30 2015

More terms from Emeric Deutsch, Jul 24 2006

A299779 Triangle read by rows: T(n,k) is the total number of cliques of size k in all partitions of all positive integers <= n.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 9, 3, 1, 1, 17, 5, 2, 1, 1, 28, 9, 4, 2, 1, 1, 47, 14, 7, 3, 2, 1, 1, 73, 24, 10, 6, 3, 2, 1, 1, 114, 35, 17, 9, 5, 3, 2, 1, 1, 170, 55, 25, 14, 8, 5, 3, 2, 1, 1, 253, 80, 38, 20, 13, 7, 5, 3, 2, 1, 1, 365, 118, 55, 31, 18, 12, 7, 5, 3, 2, 1, 1, 525, 167, 80, 44, 27, 17, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Apr 04 2018

Column k gives the partial sums of the k-th column of triangle A197126.

			Triangle begins:
    1;
    2,   1;
    5,   1,  1;
    9,   3,  1,  1;
   17,   5,  2,  1,  1;
   28,   9,  4,  2,  1,  1;
   47,  14,  7,  3,  2,  1,  1;
   73,  24, 10,  6,  3,  2,  1,  1;
  114,  35, 17,  9,  5,  3,  2,  1,  1;
  170,  55, 25, 14,  8,  5,  3,  2,  1,  1;
  253,  80, 38, 20, 13,  7,  5,  3,  2,  1,  1;
  365, 118, 55, 31, 18, 12,  7,  5,  3,  2,  1,  1;
  525, 167, 80, 44, 27, 17, 11,  7,  5,  3,  2,  1,  1;
...

Alois P. Heinz, Rows n = 1..150, flattened

Column 1 gives A000097.
Row sums give A014153.
Cf. A000041, A000070, A197126, A284870.

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
    end:
T:= proc(n, k) option remember;
      b(n$2, k)[2]+`if`(nAlois P. Heinz, Apr 27 2018

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m==k, l+{0, l[[1]]}, l]][b[n-p*m, p-1, k]], {m, 0, n/p}]]];
T[n_, k_] := b[n, n, k][[2]] + If[n < k, 0, T[n-1, k]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

T(n,k) = Sum_{j=k..n} A197126(j,k).
T(2n+1,n+1) = A000041(n). - Alois P. Heinz, Apr 27 2018
Sum_{k=1..n} k * T(n,k) = A284870(n). - Alois P. Heinz, May 14 2018

A103650 G.f.: x^2/((1-x^2)^2*Product_{i>0}(1-x^i)).

Original entry on oeis.org

0, 1, 1, 4, 5, 12, 16, 31, 42, 72, 98, 155, 210, 315, 423, 610, 812, 1136, 1498, 2047, 2674, 3585, 4642, 6125, 7865, 10240, 13046, 16791, 21237, 27060, 33993, 42933, 53591, 67155, 83332, 103687, 127956, 158196, 194217, 238720, 291663, 356582

Offset: 1

Vladeta Jovovic, Mar 26 2005

Let pi be a partition of n and b(pi,k) = Sum p, where p runs over all distinct parts p of pi whose multiplicities are >=k. Let T(n,k) = Sum b(pi,k), when pi runs over all partitions pi of n. G.f. for T(n,k) is x^k/((1-x^k)^2*Product_{i>0}(1-x^i)). a(n) = T(n,2).

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

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A014153.

Drop[ CoefficientList[ Series[ x^2/((1 - x^2)^2*Product[(1 - x^i), {i, 50}]), {x, 0, 42}], x], 1] (* Robert G. Wilson v, Mar 29 2005 *)
Table[Sum[PartitionsP[k]*(n-k)*(1 + (-1)^(n-k))/4, {k, 0, n}], {n, 1, 50}] (* Vaclav Kotesovec, Jul 30 2016 *)

a(n) = Sum_{k>0} k * A264404(n,k). - Alois P. Heinz, Nov 29 2015
For n>2, a(n) is the Euler transform of [1,3,1,1,1,1,...]. - Benedict W. J. Irwin, Jul 29 2016
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - Vaclav Kotesovec, Jul 30 2016

More terms from Robert G. Wilson v, Mar 29 2005

A116930 Sum of parts, counted without multiplicities, in all partitions of n into odd parts.

Original entry on oeis.org

1, 1, 4, 5, 10, 14, 22, 31, 44, 61, 82, 111, 145, 191, 245, 316, 399, 506, 631, 788, 973, 1200, 1468, 1792, 2174, 2630, 3167, 3802, 4547, 5422, 6445, 7638, 9029, 10642, 12515, 14679, 17181, 20061, 23379, 27185, 31554, 36551, 42268, 48787, 56224, 64681, 74300

Offset: 1

Emeric Deutsch, Feb 27 2006

nonn

			a(5) = 10 because the partitions of 5 into odd parts are [5], [3,1,1] and [1,1,1,1,1], with sum of the parts, counted without multiplicities 5 + (3+1) + 1 = 10.
a(5) = 10: There are three partitions of 5 into distinct parts, namely [5], [4,1], and [3,2]. We have phi(5) + phi(4) + phi(1) + phi(3) + phi(2) = 4 + 2 + 1 + 2 + 1 = 10. - _Peter Bala_, Dec 26 2013

Alois P. Heinz, Table of n, a(n) for n = 1..7500

Cf. A000010, A116929, A014153.

f:=x*(1+x^2)/(1-x^2)^2/product(1-x^(2*j-1),j=1..40): fser:=series(f,x=0,55): seq(coeff(fser,x^n),n=1..49);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(n>i*(i+1)/2, 0, b(n, i-1)+(p-> p+[0, p[1]
       *numtheory[phi](i)])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 15 2021

b[n_, i_] := b[n, i] = If[n == 0, {1, 0},
     If[n > i (i + 1)/2, {0, 0}, b[n, i - 1] +
     With[{p = b[n - i, Min[n-i, i-1]]}, p + {0, p[[1]]*EulerPhi[i]}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 13 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A116929(n,k).
G.f.: x(1+x^2)/[(1-x^2)^2*product(1-x^(2*j-1),j=1..infinity)].
a(n) = Sum_{parts k in all partitions of n into distinct parts} phi(k), where phi(k) is the Euler totient function (see A000010). An example is given below. - Peter Bala, Dec 26 2013
a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/3)) * n^(1/4) / (2*Pi^2). - Vaclav Kotesovec, Jun 11 2025

A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

Original entry on oeis.org

0, 1, 3, 7, 16, 28, 59, 91, 170, 269, 450, 655, 1162, 1602, 2527, 3793, 5805, 8034, 12660, 17131, 26484, 37384, 53738, 73504, 114683, 153613, 221225, 313339, 453769, 609179, 927968, 1223909, 1804710, 2522264, 3539835, 4855420, 7439870, 9765555, 14009545

Offset: 0

Alois P. Heinz, Feb 27 2013

nonn

			a(6) = 59: (1^6) + (2+1^4) + (2^2+1^2) + (2^3) + (3+1^3) + (3+2+1) + (3^2) + (4+1^2) + (4+2) + (5+1) + (6) = 1+3+5+8+4+6+9+5+6+6+6 = 59.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000070 (Sum 1), A006128 (Sum m), A014153 (Sum p), A024786 (Sum floor(1/m)), A066183 (Sum p^2*m), A066186 (Sum p*m), A073336 (Sum floor(m/p)), A116646 (Sum delta(m,2)), A117524 (Sum delta(m,3)), A103628 (Sum delta(m,1)*p), A117525 (Sum delta(m,2)*p), A197126, A213191.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^m]))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);

b[n_, p_] := b[n, p] = If[n==0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m==0, l, l+{0, l[[1]]*p^m}]][b[n-p*m, p-1]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

From Vaclav Kotesovec, May 24 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 5.0144820680945600131204662934686439430547... if mod(n,3)=0
c = 4.6144523178014379613985400559486878971522... if mod(n,3)=1
c = 4.5237761454818383598444208605033385016299... if mod(n,3)=2
(End)

A270143 a(n) = Sum_{k=0..n} (-1)^(k+1) * k * A000041(n-k).

Original entry on oeis.org

0, 1, -1, 3, -2, 6, -3, 11, -4, 19, -4, 31, -2, 50, 3, 79, 15, 122, 38, 187, 78, 284, 146, 426, 257, 635, 431, 939, 701, 1377, 1110, 2007, 1718, 2906, 2613, 4178, 3914, 5971, 5781, 8482, 8440, 11976, 12191, 16816, 17438, 23483, 24730, 32615, 34794, 45070

Offset: 0

Vaclav Kotesovec, Mar 12 2016

sign

Convolution of A000041 and A181983.

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

Cf. A000041, A014153, A087787, A270144.

Table[Sum[(-1)^(n-k+1)*PartitionsP[k]*(n-k), {k, 0, n}], {n, 0, 100}]
nmax = 100; CoefficientList[Series[x/(1 + x)^2 * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = Sum_{k=0..n} (-1)^(n-k+1) * (n-k) * A000041(k).
a(n) ~ A000041(n)/4.
a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)).
G.f.: x/(1+x)^2 * Product_{k>=1} 1/(1-x^k).

A141157 Triangle read by rows, A000012 * A140207.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 4, 3, 5, 4, 6, 6, 5, 6, 5, 8, 9, 10, 7, 7, 6, 10, 12, 15, 14, 11, 8, 7, 12, 15, 20, 21, 22, 15, 9, 8, 14, 18, 25, 28, 33, 30, 22, 10, 9, 16, 21, 30, 35, 44, 45, 44, 30, 11, 10, 18, 24, 35, 42, 55, 60, 66, 60, 42, 12, 11, 20, 27, 40, 49, 66, 75, 88, 90, 84, 56

Offset: 0

Gary W. Adamson & Roger L. Bagula, Jun 12 2008

Right border = partition numbers, A000041.

Row sums = A014153: (1, 3, 7, 14, 26, 45, 75,...).

			First few rows of the triangle are:
1;
2, 1;
3, 2, 2;
4, 3, 4, 3;
5, 4, 6, 6, 5;
6, 5, 8, 9, 10, 7;
7, 6, 10, 12, 15, 14, 11;
8, 7, 12, 15, 20, 21, 22, 15;
9, 8, 14, 18, 25, 28, 33, 30, 22;
10, 9, 16, 21, 30, 35, 44, 45, 44, 30;
...

Cf. A000041, A014153, A140207.

Triangle read by rows, A000012 * A140207; equivalent to taking partial sums of triangle A140207 terms, by columns.

a(60) split and more terms from Georg Fischer, May 29 2023

A213679 Total sum of parts <= n of multiplicity 0 in all partitions of n.

Original entry on oeis.org

0, 0, 3, 11, 36, 79, 186, 345, 672, 1163, 2026, 3273, 5388, 8301, 12912, 19349, 28961, 42071, 61253, 86921, 123404, 171972, 239020, 327386, 447743, 604255, 813645, 1084657, 1441643, 1899450, 2496510, 3255653, 4234822, 5472953, 7053217, 9038784, 11554020

Offset: 0

Alois P. Heinz, Mar 04 2013

nonn

			The partitions of n=4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4].  Parts <= 4 with multiplicity m=0 sum up to (2+3+4)+(3+4)+(1+3+4)+(2+4)+(1+2+3) = 36, thus a(4) = 36.

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

Column k=0 of A222730.

Cf. A000041, A000217, A014153.

b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0], `if`(p<1, [0$2],
      add((l->`if`(m=0, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..55);

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][[2]]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

a(n) = A000217(n)*A000041(n)-A014153(n-1).

A359279 Irregular triangle T(n,k) (n>=1, k>=1) read by rows in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive triangular numbers A000217.

Original entry on oeis.org

1, 3, 6, 1, 10, 3, 1, 15, 6, 3, 1, 1, 21, 10, 6, 3, 3, 1, 1, 28, 15, 10, 6, 6, 3, 3, 1, 1, 1, 1, 36, 21, 15, 10, 10, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1, 45, 28, 21, 15, 15, 10, 10, 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 55, 36, 28, 21, 21, 15, 15, 10, 10, 10, 10, 6, 6, 6, 6, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Dec 23 2022

All divisors of the largest partition into consecutive parts of all terms in row n are also all parts of all partitions of n.

			Triangle begins:
   1;
   3;
   6,  1;
  10,  3,  1;
  15,  6,  3,  1,  1;
  21, 10,  6,  3,  3,  1,  1;
  28, 15, 10,  6,  6,  3,  3, 1, 1, 1, 1;
  36, 21, 15, 10, 10,  6,  6, 3, 3, 3, 3, 1, 1, 1, 1;
  45, 28, 21, 15, 15, 10, 10, 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
From _Omar E. Pol_, Feb 28 2023: (Start)
For n = 4 the fourth row is [10, 3, 1]. The largest partition into consecutive parts of every term are respectively [4, 3, 2, 1], [2, 1], [1]. The divisors of these parts are [(1, 2, 4), (1, 3), (1, 2), (1)], [(1, 2), (1)], [1]. These 12 divisors are also all parts of all partitions of 4. They are  [(4), (2, 2), (3, 1), (2, 1, 1), (1, 1, 1, 1)]. (End)

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened).

Row sums give A014153 (convolution of A000041 and A000027).
This sequence has the same row sums as A176206, A299779 and A359350.
Cf. A000217, A336811, A336812.

A359279[rowmax_]:=Table[Flatten[Table[ConstantArray[(n-m)(n-m+1)/2,PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]],{n,rowmax}];
A359279[10] (* Generates 10 rows *) (* Paolo Xausa, Mar 06 2023 *)

A359279(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(m-1)-numbpart(m-2),i,(n-m+1)*(n-m+2)/2))));
A359279(10) \\ Generates 10 rows - Paolo Xausa, Mar 06 2023

T(n,k) = A000217(A336811(n,k)).

A359350 Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: row n is constructed by replacing A336811(n,k) with the largest partition into consecutive parts of A000217(A336811(n,k)).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 27 2022

All divisors of all terms in row n are also all parts of all partitions of n.
The terms of row n listed in nonincreasing order give the n-th row of A176206.
The number of k's in row n is equal to A000041(n-k), 1 <= k <= n.
The number of terms >= k in row n is equal to A000070(n-k), 1 <= k <= n.
The number of k's in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n.
The number of terms >= k in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A014153(n-k), 1 <= k <= n.
Row n is constructed replacing A336811(n,k) with the largest partition into consecutive parts of A359279(n,k).

			Triangle begins:
  1;
  2, 1;
  3, 2, 1, 1;
  4, 3, 2, 1, 2, 1, 1;
  5, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 1;
  6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 2, 1, 2, 1, 1, 1;
  ...
Or also the triangle begins:
  [1];
  [2, 1];
  [3, 2, 1],          [1];
  [4, 3, 2, 1],       [2, 1],       [1];
  [5, 4, 3, 2, 1],    [3, 2, 1],    [2, 1],    [1],    [1];
  [6, 5, 4, 3, 2, 1], [4, 3, 2, 1], [3, 2, 1], [2, 1], [2, 1], [1], [1];
  ...
For n = 3 the third row is [3, 2, 1, 1]. The divisors of these terms are [1, 3], [1, 2], [1], [1]. These six divisors are also all parts of all partitions of 3. They are [3], [2, 1], [1, 1, 1].

Paolo Xausa, Table of n, a(n) for n = 1..10980 (rows 1..21 of the triangle, flattened)

Row sums give A014153 (convolution of A000041 and A000027).
Row lengths give A000070.
Row n has A000041(n-1) blocks.
This triangle has the same row sums as A176206, A299779 and A359279.
Cf. A000217, A336811, A336812, A338156, A359280.

A359350row[n_]:=Flatten[Table[ConstantArray[Range[n-m,1,-1],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]];Array[A359350row,10] (* Paolo Xausa, Sep 01 2023 *)

cp's OEIS Frontend

A120477 Apply partial sum operator 5 times to partition numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A299779 Triangle read by rows: T(n,k) is the total number of cliques of size k in all partitions of all positive integers <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A103650 G.f.: x^2/((1-x^2)^2*Product_{i>0}(1-x^i)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A116930 Sum of parts, counted without multiplicities, in all partitions of n into odd parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A270143 a(n) = Sum_{k=0..n} (-1)^(k+1) * k * A000041(n-k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A141157 Triangle read by rows, A000012 * A140207.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A213679 Total sum of parts <= n of multiplicity 0 in all partitions of n.

Views