A322210 -id:A322210 - cp's OEIS frontend

A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k).

1, 2, 6, 14, 34, 74, 166, 350, 746, 1546, 3206, 6550, 13386, 27114, 54894, 110630, 222794, 447538, 898574, 1801590, 3610930, 7231858, 14480654, 28983246, 58003250, 116054034, 232186518, 464475166, 929116402, 1858449178, 3717247638, 7434950062, 14870628026, 29742206138, 59485920374, 118973809798, 237950730522, 475905520474

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 21 2002

nonn

See A083355 for a similar formula. - Thomas Wieder, May 07 2008
Partitions of n into 2 sorts of parts: the parts are unordered, but not the sorts; see example and formula by Wieder. - Joerg Arndt, Apr 28 2013
Convolution inverse of A070877. - George Beck, Dec 02 2018
Number of conjugacy classes of n X n matrices over GF(2).  Cf. Morrison link, section 2.9. - Geoffrey Critzer, May 26 2021

			From _Joerg Arndt_, Apr 28 2013: (Start)
There are a(3)=14 partitions of 3 with 2 ordered sorts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:1  1:0  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:1  1:0  1:0  ]
06:  [ 1:1  1:0  1:1  ]
07:  [ 1:1  1:1  1:0  ]
08:  [ 1:1  1:1  1:1  ]
09:  [ 2:0  1:0  ]
10:  [ 2:0  1:1  ]
11:  [ 2:1  1:0  ]
12:  [ 2:1  1:1  ]
13:  [ 3:0  ]
14:  [ 3:1  ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from T. D. Noe)
Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, Journal of Algebra, Vol. 309, No. 2 (2007), 654-671, arXiv:math/0609262.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Igor Pak, Greta Panova, and Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018.
N. J. A. Sloane, Transforms.

Cf. A006951, A000041, A070877.
Cf. A083355.
Column k=2 of A246935.
Cf. A048651.
Row sums of A256193.
Antidiagonal sums of A322210.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-2*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014

CoefficientList[ Series[ Product[1 / (1 - 2t^k), {k, 1, 35}], {t, 0, 35}], t]
CoefficientList[Series[E^Sum[2^k*x^k / (k*(1-x^k)), {k,1,30}],{x,0,30}],x] (* Vaclav Kotesovec, Sep 09 2014 *)
(O[x]^20 - 1/QPochhammer[2,x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

N=66; q='q+O('q^N); Vec(1/sum(n=0, N, (-2)^n*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

a(n) = (1/n)*Sum_{k=1..n} A054598(k)*a(n-k). - Vladeta Jovovic, Nov 23 2002
a(n) is asymptotic to c*2^n where c=3.46253527447396564949732... - Benoit Cloitre, Oct 26 2003. Right value of this constant is c = 1/A048651 = 3.46274661945506361153795734292443116454075790290443839132935303175891543974042... . - Vaclav Kotesovec, Sep 09 2014
Euler transform of A000031(n). - Vladeta Jovovic, Jun 23 2004
a(n) = Sum_{k=1..n} p(n,k)*A000079(k) where p(n,k) = number of integer partitions of n into k parts. - Thomas Wieder, May 07 2008
a(n) = S(n,1), where S(n,m) = 2 + Sum_{k=m..floor(n/2)} 2*S(n-k,k), S(n,n)=2, S(0,m)=1, S(n,m)=0 for n < m. - Vladimir Kruchinin, Sep 07 2014
a(n) = Sum_{lambda,mu,nu} (c^{lambda}{mu,nu})^2, where lambda ranges over all partitions of n, mu and nu range over all partitions satisfying |mu| + |nu| = n, and c^{lambda}{mu,nu} denotes a Littlewood-Richardson coefficient. - Richard Stanley, Nov 16 2014
G.f.: Sum_{i>=0} 2^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018
G.f.: Product_{j>=1} Product_{i>=1} 1/(1-x^(i*j))^A001037(j) given in Morrison link section 2.9. - Geoffrey Critzer, May 26 2021

Edited and extended by Robert G. Wilson v, May 25 2002

A322200 L.g.f.: L(x,y) = log( Product_{n>=1} 1/(1 - (x^n + y^n)) ), where L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k / (n+k) such that L(0,0) = 0, as a symmetric square table of coefficients T(n,k) read by antidiagonals starting with T(0,0) = 0.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 4, 3, 3, 4, 7, 4, 10, 4, 7, 6, 5, 10, 10, 5, 6, 12, 6, 21, 26, 21, 6, 12, 8, 7, 21, 35, 35, 21, 7, 8, 15, 8, 36, 56, 90, 56, 36, 8, 15, 13, 9, 36, 93, 126, 126, 93, 36, 9, 13, 18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28, 14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14, 24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24, 24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24, 31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31

Offset: 0

Paul D. Hanna, Nov 30 2018

			L.g.f.: L(x,y) = (x + y)/1 + (3*x^2 + 2*x*y + 3*y^2)/2 + (4*x^3 + 3*x^2*y + 3*x*y^2 + 4*y^3)/3 + (7*x^4 + 4*x^3*y + 10*x^2*y^2 + 4*x*y^3 + 7*y^4)/4 + (6*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 6*y^5)/5 + (12*x^6 + 6*x^5*y + 21*x^4*y^2 + 26*x^3*y^3 + 21*x^2*y^4 + 6*x*y^5 + 12*y^6)/6 + (8*x^7 + 7*x^6*y + 21*x^5*y^2 + 35*x^4*y^3 + 35*x^3*y^4 + 21*x^2*y^5 + 7*x*y^6 + 8*y^7)/7 + (15*x^8 + 8*x^7*y + 36*x^6*y^2 + 56*x^5*y^3 + 90*x^4*y^4 + 56*x^3*y^5 + 36*x^2*y^6 + 8*x*y^7 + 15*y^8)/8 + ...
such that
exp( L(x,y) ) = Product_{n>=1} 1/(1 - (x^n + y^n)), or
L(x,y) = Sum_{n>=1} -log(1 - (x^n + y^n)),
where
L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k),
in which the constant term is taken to be zero: L(0,0) = 0.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k/(n+k) in L(x,y) begins
0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...;
3, 3, 10, 10, 21, 21, 36, 36, 55, 55, 78, 78, 105, ...;
4, 4, 10, 26, 35, 56, 93, 120, 165, 232, 286, 364, ...;
7, 5, 21, 35, 90, 126, 230, 330, 537, 715, 1043, 1365, ...;
6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, 3018, ...;
12, 7, 36, 93, 230, 462, 994, 1716, 3073, 5035, 8120, ...;
8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, 19448, ...;
15, 9, 55, 165, 537, 1287, 3073, 6435, 13050, 24310, 44010, ...;
13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, 92378, ...;
18, 11, 78, 286, 1043, 3018, 8120, 19448, 44010, 92378, 185310, ...;
12, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, ...; ...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
0;
1, 1;
3, 2, 3;
4, 3, 3, 4;
7, 4, 10, 4, 7;
6, 5, 10, 10, 5, 6;
12, 6, 21, 26, 21, 6, 12;
8, 7, 21, 35, 35, 21, 7, 8;
15, 8, 36, 56, 90, 56, 36, 8, 15;
13, 9, 36, 93, 126, 126, 93, 36, 9, 13;
18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18;
12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12;
28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28;
14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14;
24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24;
24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24;
31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31; ...
where L(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n-k,k)*x^(n-k)*y^k / n.

Paul D. Hanna, Table of n, a(n) for n = 0..1890

Cf. A322210 (exp), A322201 (main diagonal), A322203, A322205, A322207, A322209.

Cf. A054598 (antidiagonal sums), A054599.

{L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{T(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}
for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

Sum_{k=0..n} T(n-k,k) = A054598(n) = Sum_{d|n} d*2^(n/d).
Sum_{k=0..n} T(n-k,k) * k/n = A054599(n) = Sum_{d|n} d*2^(n/d - 1).
Sum_{k=0..n} T(n-k,k) * 2^k = A322209(n) = [x^n] log( Product_{k>=1} 1/(1 - (2^k+1)*x^k) ) for n >= 0.
FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(0,0) = 0.
T(n,0) = sigma(n) for n > 0.
T(0,k) = sigma(k) for n > 0.
T(n,1) = n+1, for n >= 0.
T(1,k) = k+1, for k >= 0.
T(2*n,2) = T(2*n+1,2) = (n+1)*(2*n+3).
T(2,2*k) = T(2,2*k+1) = (k+1)*(2*k+3).
COLUMN GENERATING FUNCTIONS.
Row 0: log(P(x)), where P(x) = Product_{n>=1} 1/(1 - x^n).
Row 1: 1/(1-x)^2.
Row 2: (3 + x^2)/((1-x)*(1-x^2)^2).
Row 3: (4 - 4*x + 6*x^2 + 2*x^3 + x^4)/((1-x)^2*(1-x^3)^2).
Row 4: (7 - 9*x + 11*x^2 + 7*x^3 + 9*x^4 + x^5 + 5*x^6 + x^7)/((1-x)^2*(1-x^2)*(1-x^4)^2).
Row 5: (6 - 18*x + 33*x^2 - 16*x^3 + 10*x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8)/((1-x)^3*(1-x^5)^2).
Row 6: (12 - 41*x + 56*x^2 + 13*x^3 - 49*x^4 - 20*x^5 + 105*x^6 - 126*x^7 + 85*x^8 - 62*x^9 + 24*x^10 - 28*x^11 + 39*x^12 - 25*x^13 + 15*x^14 + x^15 + x^16) / ((1-x)^4*(1-x^2)^2*(1-x^3)*(1-x^6)^2).

A093695 Number of one-element transitions among partitions of the integer n for unlabeled parts.

Original entry on oeis.org

0, 0, 2, 4, 10, 18, 34, 56, 94, 146, 228, 340, 506, 730, 1050, 1476, 2066, 2844, 3896, 5268, 7090, 9442, 12518, 16454, 21534, 27980, 36210, 46572, 59674, 76056, 96594, 122106, 153852, 193048, 241492, 300974, 374038, 463286, 572304, 704826, 865874, 1060766

Offset: 0

Thomas Wieder, Apr 10 2004

nonn

It appears that a(n) = 2 * A000097(n-2). - George Beck, Sep 05 2014
This was proved as noted at A000097. - George Beck, Jan 11 2025
It appears that a(n) = A135348(n+1) - A000070(n). - Thomas Baruchel, May 12 2018

			In the unlabeled case we have 10 one-element transitions among all partitions of n=4: [1,1,1,1] -> [1,1,2]; [1,1,2] -> [2,2]; [1,1,2] -> [1,3]; [2,2] -> [1,3]; [1,3] -> [4] and [1,1,2] -> [1,1,1,1]; [2,2] -> [1,1,2]; [1,3] -> [1,1,2]; [1,3] -> [2,2]; [4] -> [1,3].
n=5:
partition number p=1 is [1,1,1,1,1], parts d(1,1)=1, d(2,1)=1 contribute 1;
partition number p=2 is [1,1,1,2], parts d(1,1)=1, d(2,2)=1 contribute 1, parts d(1,2)=2, d(4,2)=2 contribute 1;
partition number p=3 is [1,2,2], parts d(1,3)=1, d(2,3)=2 contribute 1, parts d(2,3)=2, d(3,3)=2 contribute 1;
partition number p=4 is [1,1,3], parts d(1,4)=1, d(2,4)=1 contribute 1, parts d(1,4)=1, d(3,4)=3 contribute 1;
partition number p=5 is [2,3], parts d(1,5)=2, d(2,5)=3 contribute 1;
partition number p=6 is [1,4], parts d(1,6)=1, d(2,6)=4 contribute 1;
partition number p=7 is [5], parts d(1,7)=5 contributes 0;
==> a(5)=2*9=18 (factor 2 if we accept up and down transitions).
a(5) = 18 because the 11 partitions of n=5+1=6 have the following sets of parts:
{1} contributes 0, {1, 2} contributes 2, {1, 2} contributes 2,
{2} contributes 0, {1, 3} contributes 2, {1, 2, 3} contributes 6,
{3} contributes 0, {1, 4} contributes 2, {2, 4} contributes 2,
{1, 5} contributes 2, {6} contributes 0,
which gives 0 + 2 + 2 + 0 + 2 + 6 + 0 + 2 + 2 + 2 + 0 = 18.
G.f. = 2*x^2 + 4*x^3 + 10*x^4 + 18*x^5 + 34*x^6 + 56*x^7 + 94*x^8 + ...

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

Cf. A093694, A089378.
Cf. A094533.
Column k=2 of triangle A322210.

A093695 := proc(n::integer) local a,ndxp,ListOfPartitions,APartition,PartOfAPartition,SetOfParts, iverbose; with(combinat): iverbose:=1; ListOfPartitions:=partition(n+1); a:=0; for ndxp from 1 to nops(ListOfPartitions) do APartition := ListOfPartitions[ndxp]; SetOfParts := convert(APartition,set); a := a + nops(SetOfParts)^2 - nops(SetOfParts); if iverbose = 1 then print ("ndxp, SetOfParts, nops(SetOfParts)^2 - nops(SetOfParts): ", ndxp,SetOfParts,nops(SetOfParts)^2 - nops(SetOfParts)); fi; # End of do-loop *** ndxp ***. end do; print("n, a(n):",n,a); end proc;
# second Maple program
b:= proc(n, i) option remember; local j, f, g;
      if n=0 then [0]
    elif i=1 then [1]
    else f:= b(n, i-1);
         for j to floor(n/i) do f:= zip((x, y)-> x+y,
             f, `if`(n=i*j, [1], [0, b(n-i*j, i-1)[]]), 0)
         od; f
      fi
    end:
a:= n-> (l-> add(i*(i-1)*l[i], i=1..nops(l)))(b(n+1, n+1)):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 05 2012

a[n_] := Block[{p = IntegerPartitions[n + 1], l = PartitionsP[n + 1]}, Sum[ Length[ Union[ p[[k]]]]^2 - Length[ Union[ p[[k]] ]], {k, l}]]; Table[ a[n], {n, 0, 40}] (* Robert G. Wilson v, Jul 13 2004, updated by Jean-François Alcover, Jan 29 2014 *)

a(n) = Sum_p=1^P(n) Sum_i=1^D(p) Sum_j=1^D(p) 1 [subject to: i <> j and d(i,p) <= d(j,p) and d(i,p) <> d(i-1,p) (if i > 1) and d(j,i) <> d(j-1,i) (if j > 1 and if d(j-1,p) has given a contribution to the sum) ]; P(n) = number of partitions of n, D(p) = number of parts in partition p, d(i,p) and d(j,p) = parts number i and j in partition p of integer n.
See the corresponding formula for a(n) for the labeled case A094533.
a(n) = Sum_i=1^P(n+1) S(i, n+1)^2 - S(i, n+1), where P(n+1) is the number of integer partitions of n+1 and S(i, n+1) is the number of parts in the set of parts of the i-th partition of n+1. (E.g. the partition [1111233] has the set of parts {1, 2, 3} and would contribute 3^2 - 3 = 6 to the sum.)
G.f.: 2*x^2 / (x^3-x^2-x+1) * Product_{m>=1} (1/(1-x^m))  (conjectured). - Thomas Baruchel, May 12 2018

More terms from Robert G. Wilson v, Jul 13 2004

A322211 a(n) = coefficient of x^n*y^n in Product_{n>=1} 1/(1 - (x^n + y^n)).

Original entry on oeis.org

1, 2, 10, 38, 158, 602, 2382, 9142, 35492, 136936, 530404, 2053848, 7972272, 30977742, 120576112, 469915012, 1833813534, 7164469910, 28021000340, 109699469798, 429850240742, 1685728936622, 6615913739206, 25983523253950, 102115250446680, 401557335718522, 1579978592844064, 6219928993470190, 24498287876663618, 96535916978924934, 380568644820360668

Offset: 0

Paul D. Hanna, Nov 30 2018

nonn

Number of subsets of partitions of 2n that have sum n. Olivier Gérard, May 07 2020

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 38*x^3 + 158*x^4 + 602*x^5 + 2382*x^6 + 9142*x^7 + 35492*x^8 + 136936*x^9 + 530404*x^10 + 2053848*x^11 + 7972272*x^12 + ...
RELATED SERIES.
The product P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)) begins
P(x,y) = 1 + (x + y) + (2*x^2 + 2*x*y + 2*y^2) + (3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3) + (5*x^4 + 7*x^3*y + 10*x^2*y^2 + 7*x*y^3 + 5*y^4) + (7*x^5 + 12*x^4*y + 18*x^3*y^2 + 18*x^2*y^3 + 12*x*y^4 + 7*y^5) + (11*x^6 + 19*x^5*y + 34*x^4*y^2 + 38*x^3*y^3 + 34*x^2*y^4 + 19*x*y^5 + 11*y^6) + (15*x^7 + 30*x^6*y + 56*x^5*y^2 + 74*x^4*y^3 + 74*x^3*y^4 + 56*x^2*y^5 + 30*x*y^6 + 15*y^7) + (22*x^8 + 45*x^7*y + 94*x^6*y^2 + 133*x^5*y^3 + 158*x^4*y^4 + 133*x^3*y^5 + 94*x^2*y^6 + 45*x*y^7 + 22*y^8) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.
The logarithm of the g.f. begins
log( A(x) ) = 2*x + 16*x^2/2 + 62*x^3/3 + 272*x^4/4 + 922*x^5/5 + 3640*x^6/6 + 12966*x^7/7 + 49872*x^8/8 + 190340*x^9/9 + 745316*x^10/10 + 2928136*x^11/11 + 11602184*x^12/12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..500 (previous b-file of terms 0..50 supplied by Vaclav Kotesovec).

Cf. A108796, A258471, A322210, A322213, A322198.

nmax = 20; s = Series[Product[1/(1 - (x^k + y^k)), {k, 1, nmax}], {x, 0, nmax}, {y, 0, nmax}]; Flatten[{1, Table[Coefficient[s, x^n*y^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Dec 04 2018 *)

{P = 1/prod(n=1,61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{a(n) = polcoeff( polcoeff( P,n,x),n,y)}
for(n=0,35, print1( a(n),", ") )

Main diagonal of square table A322210.

a(n) ~ c * 4^n / sqrt(Pi*n), where c = 1 / A048651 = 1 / Product_{k>=1} (1 - 1/2^k) = 3.46274661945506361153795734292443116454075790290443839... - Vaclav Kotesovec, Dec 23 2018

A284593 Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 2, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 2, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 6, 5, 6, 5, 6, 4, 5, 10, 12, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 12, 15, 7, 6, 8, 7, 8, 8, 8, 7, 8, 6, 7, 15

Offset: 0

Peter Bala, Mar 30 2017

Compare with A284592.

			Square array begins
  n\k| 0  1  2  3  4  5  6   7   8   9  10  11  12  13
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0  | 1  1  1  2  2  3  4   5   6   8  10  12  15  18: A000009
  1  | 1  0  1  1  1  2  2   3   3   5   5   7   8  10: A096765
  2  | 1  1  0  1  2  2  2   3   4   5   6   7   9  11: A015744
  3  | 2  1  1  2  2  3  4   6   6   8   9  12  15  18
  4  | 2  1  2  2  2  3  5   5   7   9  10  14  15  19
  5  | 3  2  2  3  3  6  6   8   9  12  16  19  22  28
  6  | 4  2  2  4  5  6  8   9  11  16  18  22  27  33
  7  | 5  3  3  6  5  8  9  14  16  20  23  29  34  41
  ...
T(3,7) = 6: the six pairs of partitions of 3 and 7 into distinct parts and with no parts in common are (3, 7), (3, 6 + 1), (3, 5 + 2), (3, 4 + 2 + 1), (2 + 1, 7) and (2 + 1, 4 + 3).

Alois P. Heinz, Antidiagonals n = 0..200, flattened
H. S. Wilf, Lectures on Integer Partitions

Rows n=0..2 give A000009, A096765, A015744.
Main diagonal gives A365662.
Antidiagonal sums give A032302.
Cf. A284592, A322210.

# A284593 as a square array
ser := taylor(taylor(mul(1 + x^j + y^j, j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand((x^i+1)*b(n-i, min(n-i, i-1)))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Aug 24 2019

nmax = 12; M = CoefficientList[#, y][[;; nmax+1]]& /@ (Product[1 + x^j + y^j, {j, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& // Expand);
T[n_, k_] := M[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

O.g.f. Product_{j >= 1} (1 + x^j + y^j) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A032302.

A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

Original entry on oeis.org

1, 3, 14, 51, 195, 663, 2345, 7707, 25744, 82980, 267812, 846150, 2676163, 8337189, 25947281, 80053128, 246468551, 754366239, 2305139065, 7014997404, 21317567297, 64606020012, 195557995054, 590855420007, 1783577678925, 5377112705874, 16199746640340, 48763788775530, 146712079122114, 441146762285301, 1326002750336702, 3984148679940612, 11967872331787643

Offset: 0

Paul D. Hanna, Dec 01 2018

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^n + 1. - Seiichi Manyama, Apr 11 2025

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ...
such that
A(x) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).
RELATED SERIES.
log( A(x) ) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ... + A322209(n)*x^n/n + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Seiichi Manyama, Generalized Euler transform.

Cf. A322209, A322200, A322210.

Cf. A000051, A266964.

{a(n) = polcoeff( 1/prod(m=1,n, 1 - (2^m+1)*x^m +x*O(x^n)),n)}
for(n=0,30, print1(a(n),", "))

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} A322200(n-k,k) * 2^k ).
a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (2^k + 1)/3^k) = 6.49344992975096517443610066284481821741772051973643441550853873760083... - Vaclav Kotesovec, Oct 04 2020
a(n) = Sum_{k=0..n} 2^k * A322210(k,n-k). - Seiichi Manyama, Apr 11 2025

A322214 a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n))^3.

Original entry on oeis.org

1, 6, -12, -6, 6, -12, -12, 96, 24, -134, -192, 114, 736, 282, -792, -1532, -270, 1932, 2004, -96, -3654, -6910, -5532, 4836, 21500, 23454, 11850, -8216, -43998, -57744, -34424, 16716, 73506, 105500, 87432, -24474, -230028, -331626, -257616, -163250, 316434, 852450, 1130284, 1175748, 361110, -652820, -1956330, -2964180, -2922288, -1965174, 187806, 3863602, 6585672, 6996900, 6199180, 366768, -7228866, -14682152, -21063366, -19602108, -10562926, 6959976, 30061386, 50110338, 66753126, 68131632, 37666392

Offset: 0

Paul D. Hanna, Dec 04 2018

sign

Compare: Product_{n>=1} (1-x^n)^3 = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)/2).

			G.f.: A(x) = 1 + 6*x - 12*x^2 - 6*x^3 + 6*x^4 - 12*x^5 - 12*x^6 + 96*x^7 + 24*x^8 - 134*x^9 - 192*x^10 + 114*x^11 + 736*x^12 + 282*x^13 - 792*x^14 - 1532*x^15 - 270*x^16 + 1932*x^17 + 2004*x^18 + ...
RELATED SERIES.
The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n))^3 begins
P(x,y) = 1 + (-3*x - 3*y) + (0*x^2 + 6*x*y + 0*y^2) + (5*x^3 + 6*x^2*y + 6*x*y^2 + 5*y^3) + (0*x^4 - 9*x^3*y - 12*x^2*y^2 - 9*x*y^3 + 0*y^4) + (0*x^5 - 9*x^4*y - 6*x^3*y^2 - 6*x^2*y^3 - 9*x*y^4 + 0*y^5) + (-7*x^6 - 9*x^5*y + 6*x^4*y^2 - 6*x^3*y^3 + 6*x^2*y^4 - 9*x*y^5 - 7*y^6) + (0*x^7 + 12*x^6*y + 12*x^5*y^2 + 27*x^4*y^3 + 27*x^3*y^4 + 12*x^2*y^5 + 12*x*y^6 + 0*y^7) + (0*x^8 + 12*x^7*y + 24*x^6*y^2 + 30*x^5*y^3 + 6*x^4*y^4 + 30*x^3*y^5 + 24*x^2*y^6 + 12*x*y^7 + 0*y^8) + (0*x^9 + 12*x^8*y - 12*x^7*y^2 - 23*x^6*y^3 - 24*x^5*y^4 - 24*x^4*y^5 - 23*x^3*y^6 - 12*x^2*y^7 + 12*x*y^8 + 0*y^9) + (9*x^10 + 12*x^9*y + 0*x^8*y^2 + 3*x^7*y^3 - 15*x^6*y^4 - 12*x^5*y^5 - 15*x^4*y^6 + 3*x^3*y^7 + 0*x^2*y^8 + 12*x*y^9 + 9*x^0*y^10) + (0*x^11 - 15*x^10*y - 36*x^9*y^2 - 54*x^8*y^3 - 60*x^7*y^4 - 60*x^6*y^5 - 60*x^5*y^6 - 60*x^4*y^7 - 54*x^3*y^8 - 36*x^2*y^9 - 15*x*y^10 + 0*y^11) + (0*x^12 - 15*x^11*y - 24*x^10*y^2 - 23*x^9*y^3 - 30*x^8*y^4 - 9*x^7*y^5 - 12*x^6*y^6 - 9*x^5*y^7 - 30*x^4*y^8 - 23*x^3*y^9 - 24*x^2*y^10 - 15*x*y^11 + 0*y^12) + (0*x^13 - 15*x^12*y - 6*x^11*y^2 - 12*x^10*y^3 + 51*x^9*y^4 + 57*x^8*y^5 + 54*x^7*y^6 + 54*x^6*y^7 + 57*x^5*y^8 + 51*x^4*y^9 - 12*x^3*y^10 - 6*x^2*y^11 - 15*x*y^12 + 0*y^13) + (0*x^14 - 15*x^13*y + 6*x^12*y^2 + 24*x^11*y^3 + 66*x^10*y^4 + 33*x^9*y^5 + 69*x^8*y^6 + 96*x^7*y^7 + 69*x^6*y^8 + 33*x^5*y^9 + 66*x^4*y^10 + 24*x^3*y^11 + 6*x^2*y^12 - 15*x*y^13 + 0*y^14) + (-11*x^15 - 15*x^14*y + 24*x^13*y^2 + 49*x^12*y^3 + 87*x^11*y^4 + 69*x^10*y^5 + 127*x^9*y^6 + 93*x^8*y^7 + 93*x^7*y^8 + 127*x^6*y^9 + 69*x^5*y^10 + 87*x^4*y^11 + 49*x^3*y^12 + 24*x^2*y^13 - 15*x*y^14 - 11*y^15) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A322210, A322211, A322213.

{P = prod(n=1, 121, (1 - (x^n + y^n) +O(x^121) +O(y^121))^3 ); }
{a(n) = polcoeff( polcoeff( P, n, x), n, y)}
for(n=0, 120, print1( a(n), ", ") )

A382956 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{p prime} 1/(1 - x^p - y^p).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 3, 2, 3, 0, 2, 3, 0, 3, 1, 1, 3, 0, 3, 3, 0, 5, 3, 6, 3, 5, 0, 3, 4, 0, 6, 4, 4, 4, 4, 6, 0, 4, 5, 0, 8, 4, 11, 8, 11, 4, 8, 0, 5, 6, 0, 10, 6, 10, 9, 9, 10, 6, 10, 0, 6, 7, 0, 13, 8, 19, 13, 28, 13, 19, 8, 13, 0, 7

Offset: 0

Seiichi Manyama, Apr 10 2025

			Square array begins:
  1, 0,  1, 1,  1,  2,  2,  3,   3,   4, ...
  0, 0,  0, 0,  0,  0,  0,  0,   0,   0, ...
  1, 0,  2, 1,  3,  3,  5,  6,   8,  10, ...
  1, 0,  1, 2,  1,  3,  4,  4,   6,   8, ...
  1, 0,  3, 1,  6,  4, 11, 10,  19,  20, ...
  2, 0,  3, 3,  4,  8,  9, 13,  17,  22, ...
  2, 0,  5, 4, 11,  9, 28, 20,  50,  50, ...
  3, 0,  6, 4, 10, 13, 20, 28,  38,  51, ...
  3, 0,  8, 6, 19, 17, 50, 38, 104,  92, ...
  4, 0, 10, 8, 20, 22, 50, 51,  92, 122, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000607, A000004.

Cf. A322210, A382955.

A(n,k) = A(k,n).

A382974 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{j>=1} 1/(1 - x^j + y^j).

Original entry on oeis.org

1, -1, 1, 0, -2, 2, -1, 2, -4, 3, 1, -3, 4, -7, 5, -1, 4, -8, 10, -12, 7, 1, -5, 14, -20, 18, -19, 11, -1, 6, -18, 34, -40, 34, -30, 15, 2, -7, 22, -51, 78, -77, 56, -45, 22, -2, 9, -30, 75, -127, 157, -139, 94, -67, 30, 2, -11, 42, -105, 196, -282, 306, -239, 146, -97, 42

Offset: 0

Seiichi Manyama, Apr 11 2025

			Square array begins:
   1,  -1,  0,   -1,   1,   -1,    1, ...
   1,  -2,  2,   -3,   4,   -5,    6, ...
   2,  -4,  4,   -8,  14,  -18,   22, ...
   3,  -7, 10,  -20,  34,  -51,   75, ...
   5, -12, 18,  -40,  78, -127,  196, ...
   7, -19, 34,  -77, 157, -282,  478, ...
  11, -30, 56, -139, 306, -582, 1048, ...

Columns k=0..1 give A000041, (-1)*A000070.
Rows n=0..1 give A081362, (-1)^k * A304631(k).
Main diagonal gives A382979.
Antidiagonal sums give A000007.
Cf. A322210.

A361286 Total over all partitions lambda of n, of factors of s_lambda in the skew Schur function s_( nu/lambda ) with (s_lambda)^2 = Sum( C(nu, lambda, lambda) s_nu ).

Original entry on oeis.org

1, 2, 6, 18, 50, 138, 430, 1242, 3666, 10938, 34598, 108098, 338634, 1058370

Offset: 0

Wouter Meeussen, Mar 07 2023

All the terms for n >= 1 so far are twice an odd integer.
In terms of Young diagrams, this counts how many original copies one gets by first adding n boxes and then removing n boxes while maintaining an allowed Young diagram shape.
Also a(n) is the total over all partitions n of the multiplicities squared, partition by partition, in the LR-expansion of (s_lambda |- n)^2. Notice that this is different from A067855 where the multipliciteis are first summed over all lambda |-n, and finally squared, then summed.

			For n=3,
    {3} -> 4 s_{3} + 2 s_{2,1}
    {2,1} -> 4 s_{3} + 10 s_{2,1} + 4 s_{1,1,1} and
    {1,1,1} -> 2 s_{2,1} + 4 s_{1,1,1}
so a(3) = 4 + 10 + 4 = 18.
Also,
s(3)^2 -> s(6)+s(3;3)+s(4;2)+s(5,1) -> {1,1,1,1} ->{1,1,1,1} ->4
  s(2;1)^2 ->s(4;2)+s(4;1;1)+s(3;3)+2 s(3;2;1)+s(3;1;1;1)+s(2;2;2)+s(2;2;1;1)
         -> {1,1,1,2,1,1,1} -> {1,1,1,4,1,1,1} -> 10
s(1;1;1)^2 -> s(2;2;2)+s(2;2;1;1)+s(2;1;1;1;1)+s(1^6) ->{1,1,1,1} ->{1,1,1,1} ->4

Cf. A067855, A322210.

(* with 'LRRule' and 'skewschur' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb *)
Tr/@ Table[Coefficient[
  Total[skewschur[#, \[Lambda], n] & /@
    LRRule[\[Lambda], \[Lambda]]], ss[\[Lambda], n] ], {n,
  13}, {\[Lambda], Partitions[n]}];
also Table[Total[
  Table[Map[Last, Tally[LRRule[\[Lambda], \[Lambda]]] ]^2, {\[Lambda],
     Partitions[n]}], 2], {n, 13}];

cp's OEIS Frontend

A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k).

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A322200 L.g.f.: L(x,y) = log( Product_{n>=1} 1/(1 - (x^n + y^n)) ), where L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k / (n+k) such that L(0,0) = 0, as a symmetric square table of coefficients T(n,k) read by antidiagonals starting with T(0,0) = 0.

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A093695 Number of one-element transitions among partitions of the integer n for unlabeled parts.

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A322211 a(n) = coefficient of x^n*y^n in Product_{n>=1} 1/(1 - (x^n + y^n)).

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A284593 Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common.

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A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

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A322214 a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n))^3.

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