A003293 -id:A003293 - cp's OEIS frontend

A323587 Number of strict (distinct parts) plane partitions of n with relatively prime parts.

1, 1, 0, 2, 2, 4, 6, 10, 12, 18, 30, 40, 48, 74, 92, 142, 172, 242, 294, 412, 490, 722, 854, 1164, 1396, 1880, 2260, 2976, 3748, 4764, 5792, 7472, 9082, 11488, 14012, 17522, 21830, 26896, 32820, 40536, 49488, 60636, 73626, 89962, 108854, 134240, 160952, 195858

Offset: 0

Gus Wiseman, Jan 20 2019

nonn

			The a(9) = 18 plane partitions:
  81   72   621   54   531   432
.
  8   7   61   62   5   51   53   42   43
  1   2   2    1    4   3    1    3    2
.
  6   5   4
  2   3   3
  1   1   2

Cf. A000219, A000837, A003293, A006951, A026007, A100883, A117433 (strict plane partitions), A300275 (plane partitions with relatively prime parts), A303546, A320802, A323584, A323585.

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]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]}],{n,30}]

Moebius transform of A117433.

A232439 Number T(n,k) of standard Young tableaux with n cells and major index k; triangle T(n,k), n>=0, 0<=k<=n*(n-1)/2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 4, 4, 4, 3, 2, 1, 1, 1, 1, 2, 4, 5, 7, 9, 9, 9, 9, 7, 5, 4, 2, 1, 1, 1, 1, 2, 4, 6, 9, 13, 16, 19, 22, 23, 23, 22, 19, 16, 13, 9, 6, 4, 2, 1, 1, 1, 1, 2, 4, 7, 10, 16, 22, 30, 37, 46, 52, 60, 62, 64, 62

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 23 2014

Rows are symmetric.
The row beginnings converge to A003293.
T(n,k) is also the number of ballot sequences of length n with k the sum of positions of all ascents, see example.

			For n=4 the 10 tableaux sorted by major index (sum of descent set) are:
:[1 2 3 4]:[1 3 4]:[1 2] [1 2 4]:[1 4] [1 2 3]:[1 3] [1 3]:[1 2]:[1]:
:         :[2]    :[3 4] [3]    :[2]   [4]    :[2]   [2 4]:[3]  :[2]:
:         :       :             :[3]          :[4]        :[4]  :[3]:
:         :       :             :             :           :     :[4]:
: ---0--- : --1-- : -----2----- : -----3----- : ----4---- : -5- : 6 :
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 1, 1, 1;
1, 1, 2, 2, 2, 1,  1;
1, 1, 2, 3, 4, 4,  4,  3,  2,  1,  1;
1, 1, 2, 4, 5, 7,  9,  9,  9,  9,  7,  5,  4,  2,  1,  1;
1, 1, 2, 4, 6, 9, 13, 16, 19, 22, 23, 23, 22, 19, 16, 13, 9, 6, 4, 2, 1, 1;
The 10 ballot sequences of length 4 are:
##   [ ballot seq] ascent positions  sum
01:  [ 1 1 1 1 ]   (none)            0
02:  [ 1 1 1 2 ]   3                 3
03:  [ 1 1 2 1 ]   2                 2
04:  [ 1 1 2 2 ]   2                 2
05:  [ 1 1 2 3 ]   2 + 3             5
06:  [ 1 2 1 1 ]   1                 1
07:  [ 1 2 1 2 ]   1 + 3             4
08:  [ 1 2 1 3 ]   1 + 3             4
09:  [ 1 2 3 1 ]   1 + 2             3
10:  [ 1 2 3 4 ]   1 + 2 + 3         6
The numbers 2, 3, and 4 appear twice, all others once, so the row four is  1, 1, 2, 2, 2, 1, 1.

Joerg Arndt and Alois P. Heinz, Rows n = 0..40, flattened
FindStat - Combinatorial Statistic Finder, The charge of the tableau, The inversion number of a standard Young tableau as defined by Haglund and Stevens, The cocharge of a standard tableau
James Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics, AMS University Lecture Series, vol. 41, 2008.
Wikipedia, Young tableau

Row sums give A000085.

Cf. A000217, A003293, A225616, A247386.

b:= proc(l, i) option remember; `if`(l=[], 1, expand(add(
      `if`(l[j]>`if`(j=1, 0, l[j-1]), `if`(j=1 and l[j]=1,
       b(subsop(1=NULL, l), j-1), b(subsop(j=l[j]-1, l), j))*
       x^`if`(j>i, add(t, t=l), 0), 0), j=1..nops(l))))
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, b([1$n, l[]], nops(l)+n),
                 add(g(n-i*j, i-1, [i$j, l[]]), j=0..n/i)):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2, [])):
seq(T(n), n=0..10);
# second Maple program (counting ballot sequences):
b:= proc(n, v, l) option remember; local w; w:=add(t, t=l);
      `if`(n<1, 1, expand(add(`if`(i=1 or l[i-1]>l[i],
      `if`(v(p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 1, [1])):
seq(T(n), n=0..10);

b[l_List, i_] := b[l, i] = If[l == {}, 1, Expand[Sum[ If[l[[j]] > If[j == 1, 0, l[[j-1]]], If[j == 1 && l[[j]] == 1, b[ReplacePart[l, 1 -> Sequence[]], j-1], b[ReplacePart[l, j -> l[[j]]-1], j]]*x^If[j>i, Total[l], 0], 0], {j, 1, Length[l]}]]] ; g[n_, i_, l_List] := g[n, i, l] = If[n == 0 || i == 1, b[Join[Array[1&, n], l], Length[l]+n], Sum[g[n-i*j, i-1, Join[Array[i&, j], l]], {j, 0, n/i}]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n, {}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 14 2015, translated from Maple *)

A275416 Triangle read by rows: T(n,k) is the number of multisets of k odd numbers with a cap of the total sum set to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 4, 3, 1, 1, 3, 8, 5, 3, 1, 1, 4, 10, 10, 5, 3, 1, 1, 4, 16, 15, 11, 5, 3, 1, 1, 5, 20, 27, 17, 11, 5, 3, 1, 1, 5, 29, 38, 32, 18, 11, 5, 3, 1, 1, 6, 35, 60, 49, 34, 18, 11, 5, 3, 1, 1, 6, 47, 84, 83, 54, 35, 18, 11, 5, 3, 1, 1, 7, 56, 122, 123

Offset: 1

R. J. Mathar, Jul 27 2016

By considering the partitions of n into k parts we set a cap on the odd numbers of each part and count the multisets (ordered k-tuples) of odd numbers where each number is not larger than the cap of its part.

Multiset transformation of A110654 or A065033.

			T(6,2) = 3+2+3 = 8 counts {1,1} {1,3}, and {3,3} from taking two odd numbers <= 3; it counts {1,1} and {1,3} from taking an odd number <= 2 and an odd number <= 4; and it counts {1,1}, {1,3} and {1,5} from taking an odd number <= 1 and an odd number <= 5.
T(6,3) = 1+2+2 = 5 counts {1,1,1} from taking three odd numbers <= 2; it counts {1,1,1} and {1,1,3} from taking an odd number <= 1 and an odd number <= 2 and an odd number <= 3; and it counts {1,1,1} and {1,1,3} from taking two odd numbers <= 1 and an odd number <= 4.
  1
  1   1
  2   1   1
  2   3   1   1
  3   4   3   1   1
  3   8   5   3   1   1
  4  10  10   5   3   1   1
  4  16  15  11   5   3   1   1
  5  20  27  17  11   5   3   1   1
  5  29  38  32  18  11   5   3   1   1
  6  35  60  49  34  18  11   5   3   1   1
  6  47  84  83  54  35  18  11   5   3   1   1
  7  56 122 123  94  56  35  18  11   5   3   1   1
  7  72 164 192 146  99  57  35  18  11   5   3   1   1

Alois P. Heinz, Rows n = 1..200, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A110654 (column 1), A003293 (row sums?), A089353 (equivalent Multiset transformation of A000027), A005232 (2nd column?), A097513 (3rd column?).

T(2n,n) gives A269628.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(ceil(i/2)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Apr 13 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[Ceiling[i/2] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

T(n,1) = A110654(n).
T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1 < k <= n.
G.f.: Product_{j>=1} (1-y*x^j)^(-ceiling(j/2)). - Alois P. Heinz, Apr 13 2017

A323451 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 9, 12, 19, 27, 39, 54, 79, 107, 150, 209, 282, 387, 525, 707, 949, 1272, 1688, 2244, 2968, 3902, 5125, 6712, 8752, 11383, 14780, 19109, 24671, 31768, 40791, 52280, 66860, 85296, 108621, 138054, 175085, 221676, 280161, 353414, 445098, 559661

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.

			The a(8) = 19 generalized Young tableaux:
  8   1 7   2 6   3 5   1 2 5   1 3 4
.
  1   2   3   1 2   1 5   1 3   1 4   2 3   1 2   1 2 3
  7   6   5   5     2     4     3     3     2 3   2
.
  1   1   1 2
  2   3   2
  5   4   3

Seiichi Manyama, Table of n, a(n) for n = 0..50
nLab, Young Diagram.
The Unapologetic Mathematician weblog, Generalized Young Tableaux.

Cf. A000085, A000219, A003293, A114736, A117433, A138178, A299968.

Cf. A323430, A323436, A323437, A323439, A323450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@sqfacs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&&And@@(UnsameQ@@@DeleteCases[Transpose[PadRight[#]],0,{2}])&]],{y,IntegerPartitions[n]}],{n,10}]

a(21)-a(45) from Seiichi Manyama, Aug 19 2020

A323580 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.

Original entry on oeis.org

1, 1, 3, 6, 13, 23, 45, 76, 136, 225, 381, 611, 1001, 1570, 2489, 3842, 5948, 9022, 13714, 20501, 30649, 45262, 66721, 97393, 141888, 204993

Offset: 0

Gus Wiseman, Jan 18 2019

			The a(5) = 23 tableaux:
  5   41   32   311   221   2111   11111
.
  1   2   11   21   11   111   111   1111
  4   3   3    2    21   2     11    1
.
  1   1   11   11   111
  1   2   1    11   1
  3   2   2    1    1
.
  1   11
  1   1
  1   1
  2   1
.
  1
  1
  1
  1
  1

nLab, Young Diagram.

Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323581.

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]]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,facs[y],{2}],And@@(GreaterEqual@@@Transpose[PadRight[#]])&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]

A005308 Bosonic string states.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 2, 4, 4, 7, 8, 14, 16, 25, 31, 47, 58, 85, 107, 153, 195, 271, 348, 480, 616, 834, 1077, 1445, 1863, 2478, 3194, 4216, 5431, 7118, 9157, 11942, 15329, 19884, 25485, 32916, 42090, 54147, 69093, 88563, 112769, 144056, 183028, 233112, 295525

Offset: 1

N. J. A. Sloane

nonn

See the reference for precise definition.

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 = 1..10000
T. Curtright, Counting symmetry patterns in the spectra of strings, in H. J. de Vega and N. Sánchez, editors, String Theory, Quantum Cosmology and Quantum Gravity. Integrable and Conformal Invariant Theories. World Scientific, Singapore, 1987, pp. 304-333.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Cf. A003293, A005986.

nmax = 50; Rest[CoefficientList[Series[x/(1-x)*Product[1/(1-x^k)^((2*k - 5 + (-1)^k)/4), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 10 2016 *)

G.f.: Product (1 - x^k)^{-c(k)}; c(k) = 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ....
Euler transform gives sequence with g.f. = x^3/((x+1)*(x-1)^2), Simon Plouffe, Master's Thesis, UQAM 1992.
a(n) ~ 2^(1/4) * exp(1/24 - 25*Pi^4/(3456*Zeta(3)) - 5*Pi^2 * n^(1/3) / (24*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2) / (A^(1/2) * sqrt(3) * Zeta(3)^(23/72) * n^(13/72)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 26 2016

A262803 Expansion of Product_{k>=1} (1 + x^k) / (1 - x^k)^k.

Original entry on oeis.org

1, 2, 5, 12, 26, 54, 110, 214, 409, 764, 1400, 2520, 4475, 7828, 13532, 23124, 39102, 65472, 108658, 178786, 291883, 472984, 761119, 1216696, 1932898, 3052462, 4793464, 7487122, 11634771, 17991760, 27692230, 42431778, 64737414, 98360742, 148853817, 224405254

Offset: 0

Vaclav Kotesovec, Oct 02 2015

nonn

Convolution of A000219 and A000009.

			G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 26*x^4 + 54*x^5 + 110*x^6 + 214*x^7 + 409*x^8 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000009, A000219, A003293, A262667.

nmax = 40; CoefficientList[Series[Product[(1+x^k)/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^-(k%2 + k), 1 + x * O(x^n)), n))}; /* Michael Somos, Oct 02 2015 */

a(n) ~ exp(1/12 - Pi^4/(1728*zeta(3)) + Pi^2 * n^(1/3)/(3*2^(7/3)*zeta(3)^(1/3)) + 3*zeta(3)^(1/3) * n^(2/3)/2^(2/3)) * zeta(3)^(7/36) / (A * sqrt(3*Pi) * 2^(29/36) * n^(25/36)), where zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
Euler transform of [ 2, 2, 4, 4, 6, 6, ...]. - Michael Somos, Oct 02 2015
G.f.: Product_{k>0} (1 - x^k)^-(k + (k mod 2)). - Michael Somos, Oct 02 2015
Convolution square of A003293. - Michael Somos, Oct 02 2015

A002985 Number of trees in an n-node wheel.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 20, 36, 64, 108, 179, 292, 464, 727, 1124, 1714, 2585, 3866, 5724, 8418, 12290, 17830, 25713, 36898, 52664, 74837, 105873, 149178, 209364, 292793, 407990, 566668, 784521, 1082848, 1490197, 2045093, 2798895, 3820629, 5202085

Offset: 1

N. J. A. Sloane

nonn

This is the number of nonequivalent spanning trees of the n-wheel graph up to isomorphism of the trees.

			All trees that span a wheel on 5 nodes are equivalent to one of the following:
      o         o         o
      |         | \     /   \
   o--o--o   o--o  o   o--o  o
      |         |           /
      o         o         o

F. Harary, P. E. O'Neil, R. C. Read and A. J. Schwenk, The number of trees in a wheel, in D. J. A. Welsh and D. R. Woodall, editors, Combinatorics. Institute of Mathematics and Its Applications. Southend-on-Sea, England, 1972, pp. 155-163.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Andrew Howroyd, Derivation of formula.
Eric Weisstein's World of Mathematics, Wheel Graph.
Index entries for sequences related to trees

Cf. A003293, A004146, A008804.

terms = 40;
A003293[n_] := SeriesCoefficient[Product[(1-x^k)^(-Ceiling[k/2]), {k, 1, terms}], {x, 0, n}];
A008804[n_] := SeriesCoefficient[1/((1-x)^4 (1+x)^2 (1+x^2)), {x, 0, n}];
a[n_] := A003293[n-1] - A008804[n-3];
Array[a, terms] (* Jean-François Alcover, Sep 02 2019 *)

\\ here b(n) is A003293 and d(n) is A008804.
b(n)={polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-ceil(k/2)), n)}
d(n)={(84+12*(-1)^n+6*I*((-I)^n-I^n)+(85+3*(-1)^n)*n+24*n^2+2*n^3)/96}
a(n)=b(n-1)-d(n-3); \\ Andrew Howroyd, Oct 09 2017

a(n) = A003293(n-1) - A008804(n-3). - Andrew Howroyd, Oct 09 2017

Terms a(32) and beyond from Andrew Howroyd, Oct 09 2017

A323581 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 14, 19, 28, 34, 48, 60, 80, 106, 134, 171, 222, 279, 354, 452, 562, 706, 884, 1100

Offset: 0

Gus Wiseman, Jan 18 2019

			The a(8) = 14 tableaux:
  8   1 7   2 6   3 5   1 2 5   1 3 4
.
  7   6   5   2 5   3 4   2 3
  1   2   3   1     1     1 2
.
  5   4
  2   3
  1   1

nLab, Young Diagram.

Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323580.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,sqfacs[y],{2}],And@@Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}]&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]

A319106 Expansion of Product_{k>=1} (1 + x^k)^ceiling(k/2).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 17, 26, 40, 60, 88, 131, 190, 276, 398, 568, 806, 1142, 1603, 2242, 3124, 4328, 5973, 8214, 11249, 15349, 20879, 28297, 38235, 51513, 69190, 92674, 123811, 164961, 219248, 290705, 384537, 507515, 668376, 878339, 1151899, 1507679, 1969503, 2567976, 3342227

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Weigh transform of 1, 1, 2, 2, 3, 3, 4, 4, ... (A110654).

N. J. A. Sloane, Transforms

Cf. A003293, A006054, A026007, A110654, A263140.

a:=series(mul((1+x^k)^ceil(k/2),k=1..100),x=0,46): seq(coeff(a,x,n),n=0..45); # Paolo P. Lava, Apr 02 2019

nmax = 45; CoefficientList[Series[Product[(1 + x^k)^Ceiling[k/2], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))(1 + x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Ceiling[d/2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

G.f.: Product_{k>=1} (1 + x^k)^A110654(k).
G.f.: Product_{k>=1} ((1 + x^(2*k-1))*(1 + x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*ceiling(d/2) ) * x^k/k).
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (2^(10/3) * 3^(4/3) * Zeta(3)^(1/3)) - Pi^4 / (2^7 * 3^4 * Zeta(3))) * Zeta(3)^(1/6) / (2^(7/8) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018

cp's OEIS Frontend

A323587 Number of strict (distinct parts) plane partitions of n with relatively prime parts.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A232439 Number T(n,k) of standard Young tableaux with n cells and major index k; triangle T(n,k), n>=0, 0<=k<=n*(n-1)/2, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A275416 Triangle read by rows: T(n,k) is the number of multisets of k odd numbers with a cap of the total sum set to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A323451 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A323580 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A005308 Bosonic string states.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A262803 Expansion of Product_{k>=1} (1 + x^k) / (1 - x^k)^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A002985 Number of trees in an n-node wheel.

Views

Author

Keywords