A101509 -id:A101509 - cp's OEIS frontend

A218482 First differences of the binomial transform of the partition numbers (A000041).

1, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812, 309274516740

Offset: 0

Paul D. Hanna, Oct 29 2012

nonn

a(n) = A103446(n) for n>=1; here a(0) is set to 1 in accordance with the definition and other important generating functions.
From Gus Wiseman, Dec 12 2022: (Start)
Also the number of sequences of compositions (A133494) with weakly decreasing lengths and total sum n. For example, the a(0) = 1 through a(3) = 8 sequences are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((12))
             ((1)(1))  ((21))
                       ((111))
                       ((1)(2))
                       ((2)(1))
                       ((11)(1))
                       ((1)(1)(1))
The case of constant lengths is A101509.
The case of strictly decreasing lengths is A129519.
The case of sequences of partitions is A141199.
The case of twice-partitions is A358831.
(End)

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...
The g.f. equals the product:
A(x) = (1-x)/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *...
and also equals the series:
A(x) = 1  +  x*(1-x)/((1-x)-x)^2  +  x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2  +  x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2  +  x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...

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

Cf. A000041, A000219, A011782, A055887, A063834, A075900, A098407, A101509, A103446, A129519, A141199, A218481.

b:= proc(n) option remember;
      add(combinat[numbpart](k)*binomial(n,k), k=0..n)
    end:
a:= n-> b(n)-b(n-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 19 2014

Flatten[{1, Table[Sum[Binomial[n-1,k]*PartitionsP[k+1],{k,0,n-1}],{n,1,30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)

{a(n)=sum(k=0,n,(binomial(n,k)-if(n>0,binomial(n-1,k)))*numbpart(k))}
for(n=0,40,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n));polcoeff(prod(k=1,n,(1-x)^k/((1-x)^k-X^k)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,x^m*(1-x)^(m*(m-1)/2)/prod(k=1,m,((1-x)^k - X^k))),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,x^(m^2)*(1-X)^m/prod(k=1,m,((1-x)^k - x^k)^2)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(exp(sum(m=1,n+1,x^m/((1-x)^m-X^m)/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(exp(sum(m=1,n+1,sigma(m)*x^m/(1-X)^m/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(prod(k=1,n,(1 + x^k/(1-X)^k)^valuation(2*k,2)),n)}

G.f.: Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
G.f.: Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).
G.f.: Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
G.f.: exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
G.f.: exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203).
G.f.: Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 34, 54, 74, 109, 146, 211, 276, 381, 501, 675, 871, 1156, 1477, 1926, 2447, 3142, 3957, 5038, 6291, 7918, 9839, 12277, 15148, 18773, 23027, 28333, 34587, 42284, 51357, 62466, 75503, 91344, 109971, 132421, 158755, 190365, 227354, 271511

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(5) = 14 split partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

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

Cf. A000005, A000219, A008284, A101509, A316245, A319066, A323295, A323300, A323307, A323429, A323434.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
      [tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
     DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 21 2022

my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.
From Seiichi Manyama, Jan 21 2022: (Start)
G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).
G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)
a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - Ridouane Oudra, Apr 13 2023

A323429 Number of rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 58, 81, 124, 169, 257, 345, 501, 684, 968, 1304, 1830, 2452, 3387, 4541, 6188, 8257, 11193, 14865, 19968, 26481, 35341, 46674, 62007, 81611, 107860, 141602, 186292, 243800, 319610, 416984, 544601, 708690, 922472, 1197018, 1553442

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are weakly decreasing.

			The a(5) = 14 matrices:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323430, A323431, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A074854 a(n) = Sum_{d|n} (2^(n-d)).

Original entry on oeis.org

1, 3, 5, 13, 17, 57, 65, 209, 321, 801, 1025, 3905, 4097, 12417, 21505, 53505, 65537, 233985, 262145, 885761, 1327105, 3147777, 4194305, 16060417, 17825793, 50339841, 84148225, 220217345, 268435457, 990937089, 1073741825, 3506503681

Offset: 1

Miklos Kristof, Sep 11 2002

A034729 = Sum_{d|n} (2^(d-1)).
A055895 = 2*A034729.
If p is a prime, then a(p) = A034729(p) = 2^(p-1)+1.
From Gus Wiseman, Jul 14 2020: (Start)
Number of ways to tile a rectangle of size n using horizontal strips. Also the number of ways to choose a composition of each part of a constant partition of n. The a(0) = 1 through a(5) = 17 splittings are:
  ()  (1)  (2)      (3)          (4)              (5)
           (1,1)    (1,2)        (1,3)            (1,4)
           (1),(1)  (2,1)        (2,2)            (2,3)
                    (1,1,1)      (3,1)            (3,2)
                    (1),(1),(1)  (1,1,2)          (4,1)
                                 (1,2,1)          (1,1,3)
                                 (2,1,1)          (1,2,2)
                                 (2),(2)          (1,3,1)
                                 (1,1,1,1)        (2,1,2)
                                 (1,1),(2)        (2,2,1)
                                 (2),(1,1)        (3,1,1)
                                 (1,1),(1,1)      (1,1,1,2)
                                 (1),(1),(1),(1)  (1,1,2,1)
                                                  (1,2,1,1)
                                                  (2,1,1,1)
                                                  (1,1,1,1,1)
                                                  (1),(1),(1),(1),(1)
(End)

			Divisors of 6 = 1,2,3,6 and 6-1 = 5, 6-2 = 4, 6-3 = 3, 6-6 = 0. a(6) = 2^5 + 2^4 + 2^3 + 2^0 = 32 + 16 + 8 + 1 = 57.
G.f. = x + 3*x^2 + 5*x^3 + 13*x^4 + 17*x^5 + 57*x^6 + 65*x^7 + ...
a(14) = 1 + 2^7 + 2^12 + 2^13 = 12417. - _Gus Wiseman_, Jun 20 2018

Gus Wiseman, Table of n, a(n) for n = 1..32

Cf. A055895, A034729.
Cf. A080267.
Cf. A051731.
The version looking at lengths instead of sums is A101509.
The strictly increasing (or strictly decreasing) version is A304961.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Requiring distinct instead of equal sums gives A336127.
Starting with a strict composition gives A336130.
Partitions of partitions are A001970.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Cf. A006951, A063834, A075900, A279787, A305551, A316245, A323433, A336128.

a[ n_] := If[ n < 1, 0, Sum[ 2^(n - d), {d, Divisors[n]}]] (* Michael Somos, Mar 28 2013 *)

a(n)=if(n<1,0,2^n*polcoeff(sum(k=1,n,2/(2-x^k),x*O(x^n)),n))

a(n) = sumdiv(n,d, 2^(n-d) ); /* Joerg Arndt, Mar 28 2013 */

G.f.: 2^n times coefficient of x^n in Sum_{k>=1} x^k/(2-x^k). - Benoit Cloitre, Apr 21 2003; corrected by Joerg Arndt, Mar 28 2013
G.f.: Sum_{k>0} 2^(k-1)*x^k/(1-2^(k-1)*x^k). - Vladeta Jovovic, Jun 24 2003
G.f.: Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=2*x and a=1/2. - Joerg Arndt, Jan 30 2011
Triangle A051731 mod 2 converted to decimal. - Philippe Deléham, Oct 04 2003
G.f.: Sum_{k>0} 1 / (2 / (2*x)^k - 1). - Michael Somos, Mar 28 2013

a(14) corrected from 9407 to 12417 by Gus Wiseman, Jun 20 2018

A323858 Number of toroidal necklaces of positive integers summing to n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 31, 44, 90, 154, 296, 524, 1035, 1881, 3636, 6869, 13208, 25150, 48585, 93188, 180192, 347617, 673201, 1303259, 2529740, 4910708, 9549665, 18579828, 36192118, 70540863, 137620889, 268655549, 524873503, 1026068477, 2007178821, 3928564237

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (necklace) case is A008965.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. Alternatively, a toroidal necklace is a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns.

			Inequivalent representatives of the a(6) = 31 toroidal necklaces:
  6  15  24  33  114  123  132  222  1113  1122  1212  11112  111111
.
  1  2  3  11  11  12  12  111
  5  4  3  13  22  12  21  111
.
  1  1  1  2  11
  1  2  3  2  11
  4  3  2  2  11
.
  1  1  1
  1  1  2
  1  2  1
  3  2  2
.
  1
  1
  1
  1
  2
.
  1
  1
  1
  1
  1
  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A000670, A008965, A059966, A101509, A179043, A184271.

Cf. A323859, A323861, A323865, A323866, A323870.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Join@@Table[Select[ptnmats[k],neckmatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,10}]

U(n,m,k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * subst(k, x, x^lcm(c,d))^(n*m/lcm(c, d))));
a(n)={if(n < 1, n==0, sum(i=1, n, sum(j=1, n\i, polcoef(U(i, j, x/(1-x) + O(x*x^n)), n))))} \\ Andrew Howroyd, Aug 18 2019

Terms a(18) and beyond from Andrew Howroyd, Aug 18 2019

A089299 Number of square plane partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 41, 57, 78, 108, 146, 202, 274, 375, 509, 690, 929, 1255, 1679, 2246, 2991, 3979, 5266, 6971, 9187, 12104, 15898, 20870, 27322, 35762, 46690, 60927, 79348, 103270, 134138, 174108, 225576, 291990, 377320, 487083

Offset: 0

N. J. A. Sloane, Dec 25 2003

nonn

Number of ways of writing n as a sum p(1,1) + p(1,2) + ... + p(1,k) + p(2,1) + ... + p(2,k) + ... + p(k,1) + ... + p(k,k) for some k so that in the square array {p(i,j)} the numbers are nonincreasing along rows and columns. All the p(i,j) are >= 1.

			a(7) = 5:
7 41 32 31 22
. 11 11 21 21
a(10) = 16 from {{10}}, {{3, 2}, {3, 2}}, {{3, 3}, {2, 2}}, {{3, 3}, {3, 1}}, {{4, 1}, {4, 1}}, {{4, 2}, {2, 2}}, {{4, 2}, {3, 1}}, {{4, 3}, {2, 1}}, {{4, 4}, {1, 1}}, {{5, 1}, {3, 1}}, {{5, 2}, {2, 1}}, {{5, 3}, {1, 1}}, {{6, 1}, {2, 1}}, {{6, 2}, {1, 1}}, {{7, 1}, {1, 1}}, {{2, 1, 1}, {1, 1, 1}, {1, 1, 1}}
From _Gus Wiseman_, Jan 16 2019: (Start)
The a(10) = 16 square plane partitions:
  [ten]
.
  [32] [33] [33] [41] [42] [42] [43] [44] [51] [52] [53] [61] [62] [71]
  [32] [22] [31] [41] [22] [31] [21] [11] [31] [21] [11] [21] [11] [11]
.
  [211]
  [111]
  [111]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A008763, A001970, A089292.

Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 16 2019 *)

G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..2k-1} (1-x^j)^min(j,2k-j). - Franklin T. Adams-Watters, Jun 14 2006

Corrected and extended by Wouter Meeussen, Dec 30 2003
a(21)-a(25) from John W. Layman, Jan 02 2004
More terms from Franklin T. Adams-Watters, Jun 14 2006
Name edited by Gus Wiseman, Jan 16 2019

A160399 a(n) = Sum_{k=1..n} binomial(n,k) * d(k), where d(k) = the number of positive divisors of k.

Original entry on oeis.org

1, 4, 11, 27, 62, 137, 296, 630, 1326, 2768, 5744, 11867, 24429, 50135, 102627, 209641, 427518, 870579, 1770536, 3596614, 7298397, 14796658, 29974913, 60681233, 122767148, 248232863, 501648844, 1013257334, 2045684971

Offset: 1

Leroy Quet, May 12 2009

nonn

Binomial transform of the sequence d(n) (A000005). - Emeric Deutsch, May 15 2009

Apparently the partial sums of A101509. - R. J. Mathar, May 17 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric M. Schmidt, The probability that the number of points on a complete intersection is squarefree, Rocky Mountain J. Math, Volume 47, Number 8 (2017), 2777-2796.

Cf. A000005. - Emeric Deutsch, May 15 2009

Cf. A101509, A185003.

List([1..10^3], n -> Sum([1..n], k -> Binomial(n,k) * Number(DivisorsInt(k)))); # Muniru A Asiru, Feb 04 2018

[&+[Binomial(n,k)*NumberOfDivisors(k):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Nov 12 2019

[&+[&+[Binomial(n,i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019

with(numtheory): seq(sum(binomial(n, k)*tau(k), k = 1 .. n), n = 1 .. 30); # Emeric Deutsch, May 15 2009
A160399 := proc(n) local k; add(binomial(n,k)*numtheory[tau](k),k=1..n) ; end: seq(A160399(n),n=1..40) ; # R. J. Mathar, May 17 2009

a[n_] := Sum[Binomial[n, k]*DivisorSigma[0, k], {k, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 25 2017 *)

a(n) = sum(k=1, n, binomial(n, k)*numdiv(k)); \\ Michel Marcus, Feb 25 2017

G.f.: (Sum_{k>=1} (x/(1-x))^k/(1-x^k/(1-x)^k))/(1-x). - Emeric Deutsch, May 15 2009
E.g.f.: exp(x)*Sum_{k>=1} d(k)*x^k/k!. - Ilya Gutkovskiy, Nov 26 2017
a(n) = 2^n*(log(n) + 2*gamma - log(2)) + O(2^n*n^(-1/4)). [Put alpha_n = beta_n = 1/2 in Thm. 4.2 of Schmidt.] - Eric M. Schmidt, Feb 03 2018
a(n) = Sum_{i=1..n} Sum_{j=1..n} binomial(n,i*j). - Ridouane Oudra, Nov 12 2019

More terms from Emeric Deutsch and R. J. Mathar, May 15 2009

A323867 Number of aperiodic arrays of positive integers summing to n.

Original entry on oeis.org

1, 1, 1, 5, 11, 33, 57, 157, 303, 683, 1358, 2974, 5932, 12560, 25328, 52400, 106256, 217875, 441278, 899955, 1822703, 3701401, 7491173, 15178253, 30691135, 62085846, 125435689, 253414326, 511547323, 1032427635, 2082551931, 4199956099, 8466869525, 17064777665

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional case is A000740.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(5) = 33 arrays:
  5  14  23  32  41  113  122  131  212  221  311  1112  1121  1211  2111
.
  1  2  3  4  11  11  12  21
  4  3  2  1  12  21  11  11
.
  1  1  1  2  2  3
  1  2  3  1  2  1
  3  2  1  2  1  1
.
  1  1  1  2
  1  1  2  1
  1  2  1  1
  2  1  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000670, A000740, A027375, A101509.

Cf. A323860, A323861, A323862, A323863, A323864, A323866, A323869.

List([0..30], A323867); # See A323861 for code; Andrew Howroyd, Aug 21 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Union@@Table[Select[ptnmats[k],apermatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,15}]

Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019

A306988 a(n) = Sum_{k=1..n} binomial(n,k)*phi(k), where phi is the Euler totient function.

Original entry on oeis.org

1, 3, 8, 20, 49, 117, 272, 620, 1395, 3107, 6852, 14964, 32395, 69647, 149002, 317712, 675749, 1433769, 3033444, 6396320, 13437913, 28130869, 58708304, 122239396, 254141275, 527946013, 1096312050, 2275897660, 4722500707, 9791471587, 20277706762, 41932520528

Offset: 1

Vaclav Kotesovec, Mar 18 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..3280
Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..5000
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A000010, A002088, A101509, A160399, A185003.

Partial sums of A131045.

Table[Sum[Binomial[n, k]*EulerPhi[k], {k, 1, n}], {n, 1, 40}]

a(n) ~ 3 * n * 2^n / Pi^2.

A323430 Number of rectangular plane partitions of n with strictly decreasing rows and columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 12, 16, 22, 27, 36, 44, 57, 72, 89, 110, 139, 170, 210, 261, 318, 390, 478, 581, 705, 860, 1036, 1252, 1511, 1816, 2178, 2618, 3127, 3743, 4471, 5330, 6347, 7564, 8984, 10674, 12669, 15016, 17780, 21050, 24868, 29371, 34655, 40836, 48080

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are strictly decreasing.

			The a(8) = 12 matrices:
  [8] [7 1] [6 2] [5 3] [5 2 1] [4 3 1]
.
  [7] [6] [5] [3 2]
  [1] [2] [3] [2 1]
.
  [5] [4]
  [2] [3]
  [1] [1]
The a(10) = 22 matrices:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [5 2] [4 2] [4 3]
  [1] [2] [3] [4] [2 1] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323431, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[Select[IntegerPartitions[#,{k}],UnsameQ@@#&]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

cp's OEIS Frontend

A218482 First differences of the binomial transform of the partition numbers (A000041).

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A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

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A323429 Number of rectangular plane partitions of n.

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A074854 a(n) = Sum_{d|n} (2^(n-d)).

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A323858 Number of toroidal necklaces of positive integers summing to n.

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A089299 Number of square plane partitions of n.

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