A070933 -id:A070933 - cp's OEIS frontend

A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k).

2, 8, 8, 7, 8, 8, 0, 9, 5, 0, 8, 6, 6, 0, 2, 4, 2, 1, 2, 7, 8, 8, 9, 9, 7, 2, 1, 9, 2, 9, 2, 3, 0, 7, 8, 0, 0, 8, 8, 9, 1, 1, 9, 0, 4, 8, 4, 0, 6, 8, 5, 7, 8, 4, 1, 1, 4, 7, 4, 1, 0, 6, 6, 1, 8, 4, 9, 0, 2, 2, 4, 0, 9, 0, 6, 8, 4, 7, 0, 1, 2, 5, 7, 0, 2, 4, 2, 8, 4, 3, 1, 9, 3, 3, 4, 8, 0, 7, 8, 2

Offset: 0

N. J. A. Sloane

This is the limiting probability that a large random binary matrix is nonsingular (cf. A002884).
This constant is very close to 2^(13/24) * sqrt(Pi/log(2)) / exp(Pi^2/(6*log(2))) = 0.288788095086602421278899775042039398383022429351580356839... - Vaclav Kotesovec, Aug 21 2018
This constant is irrational (see Penn link). - Paolo Xausa, Dec 09 2024

			(1/2)*(3/4)*(7/8)*(15/16)*... = 0.288788095086602421278899721929230780088911904840685784114741...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 318, 354-361.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Steven R. Finch, Digital Search Tree Constants. [Broken link]
Steven R. Finch, Digital Search Tree Constants. [From the Wayback machine]
Marvin Geiselhart, Ahmed Elkelesh, Moustafa Ebada, Sebastian Cammerer, and Stephan ten Brink, On the Automorphism Group of Polar Codes, arXiv:2101.09679 [cs.IT], 2021.
T. S. Jayram, Hellinger strikes back: a note on the multi-party information complexity of AND, LNCS 5687 (2009) 562-573.
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Michael Penn, A non-standard irrationality proof, YouTube video, 2024.
V. Arvind Rameshwar, Shreyas Jain, and Navin Kashyap, Sampling-Based Estimates of the Sizes of Constrained Subcodes of Reed-Muller Codes, arXiv:2309.08907 [cs.IT], 2023.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv preprint arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, Tree Searching.
Wikipedia, Pentagonal number theorem.
Wanchen Zhang, Yu Ning, Fei Shi, and Xiande Zhang, Extremal Maximal Entanglement, arXiv:2411.12208 [quant-ph], 2024. See p. 15.

Cf. A002884, A001318, A005327, A005329, A048652, A065446, A079555, A098844, A067080, A100220, A132019, A132020, A132026, A132038, A070933, A261584, A335764.

RealDigits[ Product[1 - 1/2^i, {i, 100}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
RealDigits[QPochhammer[1/2], 10, 100][[1]] (* Jean-François Alcover, Nov 18 2015 *)

default(realprecision, 20080); x=prodinf(k=1, -1/2^k, 1); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b048651.txt", n, " ", d)); \\ Harry J. Smith, May 07 2009

exp(-Sum_{k>0} sigma_1(k)/k*2^(-k)) = exp(-Sum_{k>0} A000203(k)/k*2^(-k)). - Hieronymus Fischer, Jul 28 2007
From Hieronymus Fischer, Aug 13 2007: (Start)
Equals lim inf Product_{k=0..floor(log_2(n))} floor(n/2^k)*2^k/n for n->oo.
Equals lim inf A098844(n)/n^(1+floor(log_2(n)))*2^(1/2*(1+floor(log_2(n)))*floor(log_2(n))) for n->oo.
Equals lim inf A098844(n)/n^(1+floor(log_2(n)))*2^A000217(floor(log_2(n))) for n->oo.
Equals lim inf A098844(n)/(n+1)^((1+log_2(n+1))/2) for n->oo.
Equals (1/2)*exp(-Sum_{n>0} 2^(-n)*Sum_{k|n} 1/(k*2^k)). (End)
Limit of A177510(n)/A000079(n-1) as n->infinity (conjecture). - Mats Granvik, Mar 27 2011
Product_{k >= 1} (1-1/2^k) = (1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
exp(Sum_{n>=1}(1/n/(1 - 2^n))) (according to Mathematica). - Mats Granvik, Sep 07 2016
(Sum_{k>0} (4^k-1)/(Product_{i=1..k} ((4^i-1)*(2*4^i-1))))*2 = 2/7 + 2/(3*7*31) + 2/(3*7*15*31*127)+2/(3*7*15*31*63*127*511) + ... (conjecture). - Werner Schulte, Dec 22 2016
Equals Sum_{k=-oo..oo} (-1)^k/2^((3*k+1)*k/2) (by Euler's pentagonal number theorem). - Amiram Eldar, Aug 13 2020
From Peter Bala, Dec 15 2020: (Start)
Constant C = Sum_{n >= 0} (-1)^n/( Product_{k = 1..n} (2^k - 1) ). The above conjectural result by Schulte follows by adding terms of this series in pairs.
C = (1/2)*Sum_{n >= 0} (-1/2)^n/( Product_{k = 1..n} (2^k - 1) ).
C = (3/8)*Sum_{n >= 0} (-1/4)^n/( Product_{k = 1..n} (2^k - 1) ).
1/C = Sum_{n >= 0} 2^(n*(n-1)/2)/( Product_{k = 1..n} (2^k - 1) ).
C = 1 - Sum_{n >= 0} (1/2)^(n+1)*Product_{k = 1..n} (1 - 1/2^k).
This latter identity generalizes as:
C = Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*C = 1 - Sum_{n >= 0} (1/8)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*C = 6 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*15*C = 91 - Sum_{n >= 0} (1/32)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
and so on, where the sequence [1, 0, 1, 6, 91, ...] is A005327.
(End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(2)) * exp(log(2)/24 - Pi^2/(6*log(2))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A005329(n).
Equals exp(-A335764). (End)
Equals 1/A065446. - Hugo Pfoertner, Nov 23 2024

Corrected by Hieronymus Fischer, Jul 28 2007

A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 4, 6, 2, 7, 2, 6, 4, 4, 4, 7, 2, 4, 4, 7, 2, 8, 2, 6, 6, 4, 2, 7, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 5, 7, 4, 8, 2, 6, 4, 7, 2, 8, 2, 4, 6, 6, 4, 8, 2, 8, 5, 4, 2, 9, 4, 4, 4

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer n is a subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Position of first appearance of n appears to be A259941(n-1) = least Heinz number of a complete partition of n-1. - Gus Wiseman, Nov 16 2023

			The subset-sums of (5,1,1,1) are {0, 1, 2, 3, 5, 6, 7, 8} so a(88) = 8.
The subset-sums of (4,3,1) are {0, 1, 3, 4, 5, 7, 8} so a(70) = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000005, A000041, A000720, A001222, A056239, A108917, A112798, A122111, A122768, A215366, A276024, A284640, A296150, A299702.
Positions of first appearances are A259941.
The triangle for this rank statistic is A365658.
The semi version is A366739, sum A366738, strict A366741.
Cf. A032302, A070933, A304792, A304793, A365543, A365920, A367095.

Table[Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}]

a(n) <= A000005(n) and a(n) = A000005(n) iff n is the Heinz number of a knapsack partition (A299702).

Comment corrected by Gus Wiseman, Aug 09 2024

A006951 Number of conjugacy classes in GL(n,2).

Original entry on oeis.org

1, 1, 3, 6, 14, 27, 60, 117, 246, 490, 1002, 1998, 4053, 8088, 16284, 32559, 65330, 130626, 261726, 523374, 1047690, 2095314, 4192479, 8384808, 16773552, 33546736, 67101273, 134202258, 268420086, 536839446, 1073710914, 2147420250, 4294904430, 8589807438

Offset: 0

N. J. A. Sloane

nonn

Unlabeled permutations of sets. - Christian G. Bower, Jan 29 2004
From Joerg Arndt, Jan 02 2013: (Start)
Set q=2 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P=[p_1^m_1, p_2^m_2, ..., p_L^m_L], see the Macdonald reference.
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power are:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.
(End)
From Gus Wiseman, Jan 21 2019: (Start)
Also the number of ways to split an integer partition of n into consecutive constant subsequences. For example, the a(5) = 27 ways (subsequences shown as rows) are:
  5   11111
.
  4   3   3    22   2     1111   1      111   11
  1   2   11   1    111   1      1111   11    111
.
  3   2   2    2    111   1     1     11   11   1
  1   2   11   1    1     111   1     11   1    11
  1   1   1    11   1     1     111   1    11   11
.
  2   11   1    1    1
  1   1    11   1    1
  1   1    1    11   1
  1   1    1    1    11
.
  1
  1
  1
  1
  1
(End)

			For the 5 partitions of 4 (namely [1^4]; [2,1^2]; [2^2]; [3,1]; [4]) we have
(f(m) = 2^(m-1)*(2-1) = 2^(m-1) and)
f([1^4]) = 2^3 = 8,
f([2,1^2]) = 1*2^1 = 2,
f([2^2]) = 2^1 = 2,
f([3,1]) = 1*1 = 1,
f([4]) = 1,
the sum is 8+2+2+1+1 = 14 = a(4).
- _Joerg Arndt_, Jan 02 2013

W. D. Smith, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 161
I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).
N. J. A. Sloane, Transforms

Cf. A006952, A049314, A049315, A049316, A070933, A264685, A264687.
Column k=0 of A218698. - Alois P. Heinz, Nov 04 2012
Cf. A100471, A100883, A279784, A279786, A323433, A323582, A323583.

/* The program does not work for n>19: */
[1] cat [NumberOfClasses(GL(n,2)): n in [1..19]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi Jan 24 2013

with(numtheory):
b:= n-> add(phi(d)*2^(n/d), d=divisors(n))/n-1:
a:= proc(n) option remember; `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 20 2012

b[n_] := Sum[EulerPhi[d]*2^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Table[Sum[2^(Length[ptn]-Length[Split[ptn]]),{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 21 2019 *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-2*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 02 2013 */

G.f.: Product_{n>=1} (1-x^n)/(1-2*x^n). - Joerg Arndt, Jan 02 2013
The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in Product_{k>=1} (1-t^k)/(1-q*t^k). - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001
Euler transform of A008965. - Christian G. Bower, Jan 29 2004
a(n) ~ 2^n - (1+sqrt(2) + (-1)^n*(1-sqrt(2))) * 2^(n/2-1). - Vaclav Kotesovec, Nov 21 2015
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(2^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

More terms from Christian G. Bower, Jan 29 2004

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0

Offset: 0

Alois P. Heinz, Sep 08 2014

In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - Vaclav Kotesovec, Mar 19 2015

When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field of size k. - Geoffrey Critzer, Nov 11 2022

			A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6,      7, ...
  0,  2,   6,   12,    20,     30,     42,     56, ...
  0,  3,  14,   39,    84,    155,    258,    399, ...
  0,  5,  34,  129,   356,    805,   1590,   2849, ...
  0,  7,  74,  399,  1444,   4055,   9582,  19999, ...
  0, 11, 166, 1245,  5876,  20455,  57786, 140441, ...
  0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Columns k=0-10 give: A000007, A000041, A070933, A242587, A246936, A246937, A246938, A246939, A246940, A246941, A246942.
Rows n=0-4 give: A000012, A001477, A002378, A027444, A186636.
Main diagonal gives A124577.
Cf. A144064, A255970, A256130.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; A[n_, k_] := b[n, n, k];  Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)

G.f. of column k: Product_{i>=1} 1/(1-k*x^i).

T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i).

A323583 Number of ways to split an integer partition of n into consecutive subsequences.

Original entry on oeis.org

1, 1, 3, 7, 17, 37, 83, 175, 373, 773, 1603, 3275, 6693, 13557, 27447, 55315, 111397, 223769, 449287, 900795, 1805465, 3615929, 7240327, 14491623, 29001625, 58027017, 116093259, 232237583, 464558201, 929224589, 1858623819, 3717475031, 7435314013, 14871103069

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

			The a(3) = 7 ways to split an integer partition of 3 into consecutive subsequences are (3), (21), (2)(1), (111), (11)(1), (1)(11), (1)(1)(1).

Cf. A006951, A070933, A100883, A279784, A279786, A323433, A323582.

b:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> ceil(b(n$2)):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 01 2023

Table[Sum[2^(Length[ptn]-1),{ptn,IntegerPartitions[n]}],{n,40}]
(* Second program: *)
(1/2) CoefficientList[1 - 1/QPochhammer[2, x] + O[x]^100 , x] (* Jean-François Alcover, Jan 02 2022, after Vladimir Reshetnikov in A070933 *)

a(n) = A070933(n)/2.
O.g.f.: (1/2)*Product_{n >= 1} 1/(1 - 2*x^n).
G.f.: 1 + Sum_{k>=1} 2^(k - 1) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020

A242587 The number of conjugacy classes of n X n matrices over F_3.

Original entry on oeis.org

1, 3, 12, 39, 129, 399, 1245, 3783, 11514, 34734, 104754, 314922, 946623, 2842077, 8532147, 25603788, 76830033, 230513439, 691598901, 2074870002, 6224790639, 18674600664, 56024355396, 168073769199, 504222998115, 1512671142432, 4538018555652, 13614062210490

Offset: 0

R. J. Mathar, May 18 2014

nonn

Apparently the Euler transform of A001867.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
K. E. Morrison, Integer sequences and matrices over finite fields, J. Int. Seq. 9 (2006) # 06.2.1, Section 1.16.

Cf. A070933 (over F_2).

Column k=3 of A246935.

A242587 := proc(n)
    local r,x ;
    if n  = 0 then
        1;
    else
        1/mul(1-3*x^r,r=1..n) ;
        convert(%,parfrac,x) ;
        coeftayl(%,x=0,n) ;
    end if;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 3*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, 3*b[n-i, i]]]] ; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 03 2015, after Alois P. Heinz *)
(O[x]^20 - 2/QPochhammer[3, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

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

G.f.: 1/Product_{r>=1} (1-3*x^r).
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, 3*S(n-k,k))+3, S(n,n)=3, S(0,m)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 07 2014
a(n) ~ c * 3^n, where c = Product_{k>=1} 1/(1-1/3^k) = 1.7853123419985341903674... . - Vaclav Kotesovec, Mar 19 2015
G.f.: Sum_{i>=0} 3^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A256193 Number T(n,k) of partitions of n into two sorts of parts having exactly k parts of the second sort; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 5, 12, 11, 5, 1, 7, 20, 24, 16, 6, 1, 11, 35, 49, 41, 22, 7, 1, 15, 54, 89, 91, 63, 29, 8, 1, 22, 86, 158, 186, 155, 92, 37, 9, 1, 30, 128, 262, 351, 342, 247, 129, 46, 10, 1, 42, 192, 428, 635, 700, 590, 376, 175, 56, 11, 1

Offset: 0

Alois P. Heinz, Mar 19 2015

			T(3,0) = 3: 111, 21, 3.
T(3,1) = 6: 1'11, 11'1, 111', 2'1, 21', 3'.
T(3,2) = 4: 1'1'1, 1'11', 11'1', 2'1'.
T(3,3) = 1: 1'1'1'.
Triangle T(n,k) begins:
   1;
   1,   1;
   2,   3,   1;
   3,   6,   4,   1;
   5,  12,  11,   5,   1;
   7,  20,  24,  16,   6,   1;
  11,  35,  49,  41,  22,   7,   1;
  15,  54,  89,  91,  63,  29,   8,   1;
  22,  86, 158, 186, 155,  92,  37,   9,  1;
  30, 128, 262, 351, 342, 247, 129,  46, 10,  1;
  42, 192, 428, 635, 700, 590, 376, 175, 56, 11,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
William Dugan, Sam Glennon, Paul E. Gunnells, and Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], Feb 2017.
Emmy Huang and Ray Tang, Minimum Decomposition on Maxmin Trees, arXiv:2310.14385 [math.CO], 2023.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
Wikipedia, q-Pochhammer symbol

Column k=0-10 gives: A000041, A006128, A258472, A258473, A258474, A258475, A258476, A258477, A258478, A258479, A258480.
T(2n,n) gives A258471.
Row sums give A070933.
Cf. A278464.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
       binomial(j, t), t=0..j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]* Sum[x^t*Binomial[j, t], {t, 0, j}], {j, 0, n/i}]]]]; T[n_] := Function[ p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
Table[SeriesCoefficient[FunctionExpand[1/QPochhammer[q + x, q, n]], {q, 0, n - k}, {x, 0, k}], {n, 0, 10}, {k, 0, n}] // Column (* Vladimir Reshetnikov, Nov 22 2016 *)

T(n,k) = [x^k] [q^(n-k)] 1/(q+x; q)inf = [x^k] [q^(n-k)] 1/(q+x; q)_n, where (x; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 22 2016

Sum_{k=0..n} k * T(n,k) = A278464(n). - Alois P. Heinz, Nov 22 2016

A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

Original entry on oeis.org

1, 2, 8, 26, 94, 326, 1196, 4358, 16248, 60854, 230184, 874878, 3343614, 12825418, 49368388, 190554410, 737328366, 2858974502, 11106267880, 43215101102, 168398785002, 657070401106, 2566847255572, 10038191414610, 39295007540748

Offset: 0

Richard Stanley, Feb 15 2002

Original name: "Squared length of sum of s_lambda^2, where s_lambda is a Schur function and lambda ranges over all partitions of n."
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4. - Seiichi Manyama, Apr 22 2018
The symbol "|-" means "is a partition of", cf. MathWorld link and the Geloun & Ramgoolam paper. The Littlewood-Richardson coefficients allow a product of two Schur functions to be expressed as a linear combination of Schur functions of the corresponding degree. (The Schur functions symmetric in all n variables correspond to Schur polynomials of partitions extended with 0's to length n.) - M. F. Hasler, Jan 19 2020
See A070933 for similar sums of squares of Littlewood-Richardson coefficients. - M. F. Hasler, Jan 20 2020

			For n=3 the s_lambda^2 summed over all partitions of n and decomposed into a sum of Schur functions yields
    s(6) + 2 s(3,3) + 2 s(4,2) + s(5,1) + 2 s(2,2,2) + 2 s(3,2,1) + s(4,1,1)
    + 2 s(2,2,1,1) + s(3,1,1,1) + s(2,1,1,1,1) + s(1,1,1,1,1,1),
  and the sum of the squares of the coefficients {1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1} gives a(3) = 26.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Eric Weisstein's World of Mathematics, Partition.
Wikipedia, Littlewood-Richardson rule, as of Dec 18 2018.
Wikipedia, Schur polynomial, as of Jan 13 2020.

Cf. A001868.
List of partitions: A036037, A080577, A181317, A330370.
Cf. A070933 (Sum_{lambda,mu,nu} (c^{lambda}_{mu,nu})^2, |mu| = |nu| = n).
Cf. A003040 (maximum number of standard tableaux of the Ferrers diagrams of the partitions of n).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      binomial(n+n, n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 24 2019

Table[Tr[(Apply[List,
  Sum[Tr[s @@@ LRRule[\[Lambda], \[Lambda]]],
   {\[Lambda], Partitions[n]}]] /. s[] -> 1)^2], {n, 1, 10}];
(* with 'LRRule' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb - Wouter Meeussen, Jan 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Binomial[n+n, n],
     Sum[b[j, 1]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 02 2022, after Alois P. Heinz *)

A067855_upto(N)=Vec(1/sqrt(prod(i=1,N-1,1-4*'x^i+O('x^N)))) \\ M. F. Hasler, Jan 23 2020

G.f.: 1/sqrt(Product_{i >= 1} (1 - 4*x^i)).
Euler transform of A001868(n)/2. a(n) = Sum_{pi} Product_{m=1..n} binomial(2*p(m), p(m)), where pi runs through all nonnegative solutions of p(1) + 2*p(2) + ... + n*p(n)=n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ 2^(2*n) / sqrt(c*Pi*n), where c = QPochhammer[1/4] = 0.688537537120339... - Vaclav Kotesovec, Apr 22 2018
By definition, a(n) = Sum_{mu |- 2n} c_mu^2 where Sum_{lambda |- n} s_lambda^2 = Sum_{mu |- 2n} c_mu s_mu, where s_lambda are the Schur polynomials (symmetric in 2n variables) and the sums run over all partitions of n resp. 2n. - M. F. Hasler, Jan 19 2020

More terms from Vladeta Jovovic, Mar 25 2006

Name edited by M. F. Hasler following observations by Wouter Meeussen, Jan 17 2020

A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).

Original entry on oeis.org

1, 4, 12, 36, 92, 228, 540, 1236, 2748, 6004, 12876, 27252, 57036, 118308, 243564, 498564, 1015484, 2060484, 4167804, 8409588, 16934748, 34049940, 68378220, 137185428, 275026476, 551052676, 1103618508, 2209525092, 4422484764, 8850120420, 17707920924

Offset: 0

Vaclav Kotesovec, Aug 25 2015

nonn

Cf. A032302, A070933, A261563.

nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1)), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 - QPochhammer[-2, x]/(3 QPochhammer[2, x]))[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = c * 2^n, where c = 1/(A048651 * A083864) = 2*Product_{j>=1} (2^j+1)/(2^j-1) = 16.5119758715565001310882816988645462530540032335764606912075051272567456...

A322210 G.f.: P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as a square table of coefficients T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 10, 7, 5, 7, 12, 18, 18, 12, 7, 11, 19, 34, 38, 34, 19, 11, 15, 30, 56, 74, 74, 56, 30, 15, 22, 45, 94, 133, 158, 133, 94, 45, 22, 30, 67, 146, 233, 297, 297, 233, 146, 67, 30, 42, 97, 228, 385, 550, 602, 550, 385, 228, 97, 42, 56, 139, 340, 623, 951, 1166, 1166, 951, 623, 340, 139, 56

Offset: 0

Paul D. Hanna, Nov 30 2018

Conjecture 1: the triangular table T(n,k) is the number of ways to form the subsum k from the partitions of n, where n and k are integers such that 0 <= k <= n. For example, t(4,2)=10; the five partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1) with subsum 2 occurring {0,0,2,2,6) times for a total of 10. - George Beck, Jan 03 2020
From Wouter Meeussen, Mar 09 2023: (Start)
Conjecture 2: the square table T(n,k) is the coefficient of s_lambda in the sum over all partitions lambda |-n and nu |-k of (s_rho/mu) where s_lambda*s_mu = Sum(rho|-n+k; C(rho, lambda, mu) s_rho). Simply stated as: multiply lambda with mu, and, for each term in the result, take the skew Schur function with mu and count how often you get the original lambda back. Sum up over all lambda and mu of the size n and k.
Conjecture 3: the triangular table T(n,k) is analogous to conjecture 2, but counting s_lambda in s_(lambda/mu) * s_mu with lambda |- n and mu |- k and 0<=k<=n. (End)

			G.f.: 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) + ...
such that
P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)),
where
P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
   1,   1,   2,    3,    5,     7,    11,     15,     22,     30, ...
   1,   2,   4,    7,   12,    19,    30,     45,     67,     97, ...
   2,   4,  10,   18,   34,    56,    94,    146,    228,    340, ...
   3,   7,  18,   38,   74,   133,   233,    385,    623,    977, ...
   5,  12,  34,   74,  158,   297,   550,    951,   1614,   2627, ...
   7,  19,  56,  133,  297,   602,  1166,   2133,   3775,   6437, ...
  11,  30,  94,  233,  550,  1166,  2382,   4551,   8424,  14953, ...
  15,  45, 146,  385,  951,  2133,  4551,   9142,  17639,  32680, ...
  22,  67, 228,  623, 1614,  3775,  8424,  17639,  35492,  68356, ...
  30,  97, 340,  977, 2627,  6437, 14953,  32680,  68356, 136936, ...
  42, 139, 506, 1501, 4202, 10692, 25835,  58659, 127443, 264747, ...
  56, 195, 730, 2255, 6531, 17290, 43313, 102149, 229998, 495195, ...
  ...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
   1;
   1,   1;
   2,   2,   2;
   3,   4,   4,   3;
   5,   7,  10,   7,    5;
   7,  12,  18,  18,   12,    7;
  11,  19,  34,  38,   34,   19,   11;
  15,  30,  56,  74,   74,   56,   30,   15;
  22,  45,  94, 133,  158,  133,   94,   45,   22;
  30,  67, 146, 233,  297,  297,  233,  146,   67,  30;
  42,  97, 228, 385,  550,  602,  550,  385,  228,  97,  42;
  56, 139, 340, 623,  951, 1166, 1166,  951,  623, 340, 139,  56;
  77, 195, 506, 977, 1614, 2133, 2382, 2133, 1614, 977, 506, 195, 77;
  ...

Alois P. Heinz, Antidiagonals n = 0..200 (first 61 antidiagonals from Paul D. Hanna)

Cf. A322200 (log).
Cf. A000041 (row 0 = partitions), A000070 (row 1), A093695(k+2) (row 2).
Main diagonal gives A322211.
Antidiagonal sums give A070933.
Cf. A284593.
Cf. A361286.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1,
      (x+1)^n, b(n, i-1) +(x^i+1)*b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 23 2019

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + (x^i + 1) b[n - i, Min[n - i, i]]]];
T[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{P = 1/prod(n=1,61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{T(n,k) = polcoeff( polcoeff( P,n,x),k,y)}
for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(n,0) = A000041(n) for n >= 0, where A000041 is the partition numbers.
T(n,1) = A000070(n) for n >= 0, where A000070 is the sum of partitions.
ROW GENERATING FUNCTIONS.
Row 0: 1/( Product_{n>=1} (1 - x^n) ).
Row 1: 1/( (1-x) * Product_{n>=1} (1 - x^n) ).
Row 2: 2/( (1-x) * (1-x^2) * Product_{n>=1} (1 - x^n) ).

cp's OEIS Frontend

A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k).

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A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

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A006951 Number of conjugacy classes in GL(n,2).

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A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A323583 Number of ways to split an integer partition of n into consecutive subsequences.

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A242587 The number of conjugacy classes of n X n matrices over F_3.

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A256193 Number T(n,k) of partitions of n into two sorts of parts having exactly k parts of the second sort; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).