A258268 -id:A258268 - cp's OEIS frontend

A097356 Number of partitions of n into parts not greater than sqrt(n).

1, 1, 1, 1, 3, 3, 4, 4, 5, 12, 14, 16, 19, 21, 24, 27, 64, 72, 84, 94, 108, 120, 136, 150, 169, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 16475, 18138, 19928, 21873, 23961

Offset: 0

Reinhard Zumkeller, Aug 08 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A194020, A000196, A000041, A097355.

a097356 n = p [1..a000196 n] n where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
a:= n-> b(n, (r-> `if`(r*r>n, r-1, r))(isqrt(n))):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 02 2018

Table[Length[IntegerPartitions[n,Floor[Sqrt[n]]]],{n,70}] (* Harvey P. Dale, May 11 2011 *)
f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; Table[ f[n, Floor[Sqrt[n]]], {n, 53}] (* Robert G. Wilson v, Aug 13 2011 *)

a(n,k=sqrtint(n))=if(min(n,k)<2,1,sum(i=1,min(k,n),a(n-i,i))) \\ Charles R Greathouse IV, Aug 12 2011

a(n^2) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.1582087202672504149766310999238... [see A206226, constant c(1)]. The upper bound of a(n) is c * d^sqrt(n) / n, see graph. For the lower bound, the constant c = 0.088154883798697116... (conjectured). - Vaclav Kotesovec, Jan 08 2024

A206226 Number of partitions of n^2 into parts not greater than n.

Original entry on oeis.org

1, 1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574, 97299665768397, 800212617435074, 6617003142869419, 54985826573015541, 458962108485797208, 3846526994743330075

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Also the number of partitions of n^2 using n or fewer numbers. Thus for n=3 one has: 9; 1,8; 2,7; 3,6; 4,5; 1,1,7; 1,2,6; 1,3,5; 1,4,4; 2,2,5; 2,3,4; 3,3,3. - J. M. Bergot, Mar 26 2014 [computations done by Charles R Greathouse IV]
The partitions in the comments above are the conjugates of the partitions in the definition. By conjugation we have: "partitions into parts <= m" are equinumerous with "partitions into at most m parts". - Joerg Arndt, Mar 31 2014
From Vaclav Kotesovec, May 25 2015: (Start)
In general, "number of partitions of j*n^2 into parts that are at most n" is (for j>0) asymptotic to c(j) * d(j)^n / n^2, where c(j) and d(j) are a constants.
-------
j  c(j)
1  0.1582087202672504149766310999238...
2  0.0794245035465730707705885572860...
3  0.0530017980244665552354063060738...
4  0.0397666338404544208556554596295...
5  0.0318193213988281353709268311928...
...
17 0.0093617308583114626385718275875...
   c(j) for big j asymptotically approaches 1 / (2*Pi*j).
---------
j    d(j)
1    9.15337019245412246194853029240... = A258268
2   16.57962120993269533568313969522...
3   23.98280768122086592445663786762...
4   31.37931997386325137074644287711...
5   38.77298550971449870728474612568...
...
17 127.45526806942537991146993713837...
   d(j) for big j asymptotically approaches j * exp(2).
(End)
d(j) = r^(2*j+1)/(r-1), where r is the root of the equation polylog(2, 1-r) + (j+1/2)*log(r)^2 = 0. - Vaclav Kotesovec, Jun 11 2015

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206227, A206240, A107379, A258268.
Column k=2 of A238016.
Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n^2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)
(* A program to compute the constants d(j) *) Table[r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->60], {j, 1, 5}] (* Vaclav Kotesovec, Jun 11 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2)))),n^2)}
for(n=0,25,print1(a(n),", "))

a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - Vaclav Kotesovec, Sep 07 2014

A206240 Number of partitions of n^2-n into parts not greater than n.

Original entry on oeis.org

1, 1, 2, 7, 34, 192, 1206, 8033, 55974, 403016, 2977866, 22464381, 172388026, 1341929845, 10573800028, 84192383755, 676491536028, 5479185281572, 44692412971566, 366844007355202, 3028143252035976, 25123376972033392, 209401287806758273, 1752674793617241002

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Also the number of partitions of n^2 into exactly n parts. - Seiichi Manyama, May 07 2018

			From _Seiichi Manyama_, May 07 2018: (Start)
n | Partitions of n^2 into exactly n parts
--+-------------------------------------------------------
1 | 1.
2 | 3+1 = 2+2.
3 | 7+1+1 = 6+2+1 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 3+3+3. (End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206226, A206227, A107379, A258268.

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2-n, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n-1)}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

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

a(n) = [x^(n^2-n)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.153370192454122461948530292401354540073... = A258268, c = 0.07005383646855329845970382163053268... . - Vaclav Kotesovec, Sep 07 2014

A258234 Decimal expansion of a constant related to A107379.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 24 2015

Limit n->infinity (Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx)^(1/n) = Limit n->infinity (A258191(n)/A258192(n))^(1/n) = 1/A258234 = 0.18515528932235959464731321119795428527382236445907508398560553036... .

			5.4008719041181541524660911191042700520294...

StackExchange - Mathematica, No response to an infinite limit

Cf. A107379, A173519, A258191, A258192, A258268, A258788.

r^2/(r-1) /.FindRoot[-PolyLog[2, 1-r] == Log[r]^2, {r, E}, WorkingPrecision->117] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A107379(n)^(1/n).

Equals limit n->infinity A173519(n)^(1/n).

More terms from Vaclav Kotesovec, Jun 09 2015

A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)).

Original entry on oeis.org

1, 1, 5, 27, 169, 1115, 7760, 55748, 411498, 3101490, 23785645, 185064559, 1457664666, 11602828475, 93205739436, 754751603157, 6155229065861, 50515624923790, 416930705579538, 3458726257239312, 28825340825747729, 241245120218823892, 2026803168946440648

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A258268, A258788, A258791, A258793, A258794.

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n

Table[SeriesCoefficient[1/Product[x^(2*k)*(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}], {x, 0, n*(n+2)}], {n, 0, 30}]

a(n) ~ c * d^n / n^2, where d = A258268 = 9.15337019245412246194853029240135454007332720412184884968926320147613... = r^3/(r-1), where r is the root of the equation polylog(2, 1-r) + 3*log(r)^2/2 = 0, c = 0.8069142856822510276258439534144172057548... .

A161407 Number of partitions of n^2 into parts smaller than n.

Original entry on oeis.org

1, 0, 1, 5, 30, 185, 1226, 8442, 60289, 442089, 3314203, 25295011, 195990980, 1538069121, 12203218743, 97746332667, 789480879664, 6423539487002, 52607252796831, 433368610079872, 3588859890833443, 29862449600982149, 249560820679038935, 2093852201126089073

Offset: 0

Reinhard Zumkeller, Jun 10 2009

nonn

			a(3) = #{2+2+2+2+1, 2+2+2+1+1+1, 2+2+5x1, 2+7x1, 9x1} = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..320

Cf. A072213, A093115, A109655, A161408.

a := proc (n) local G, Gser: G := 1/(product(1-x^j, j = 1 .. n-1)): Gser := series(G, x = 0, n^2+5): coeff(Gser, x, n^2) end proc: 1, seq(a(n), n = 1 .. 23); # Emeric Deutsch, Jun 20 2009
# 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, b(n-i, i))))
    end:
a:= n-> b(n^2, n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 21 2014

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n^2, n-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

a(n) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.0881548837986971165169272782933415... - Vaclav Kotesovec, Sep 08 2021

More terms from Emeric Deutsch, Jun 20 2009

a(0)=1 from Alois P. Heinz, Dec 21 2014

A206227 Number of partitions of n^2+n into parts not greater than n.

Original entry on oeis.org

1, 1, 4, 19, 108, 674, 4494, 31275, 225132, 1662894, 12541802, 96225037, 748935563, 5900502806, 46976736513, 377425326138, 3056671009814, 24930725879856, 204623068332997, 1688980598900228, 14012122025369431, 116784468316023069, 977437078888272796, 8212186058546599006

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206226, A206240, A107379, A258268.

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2+n, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

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

a(n) = [x^(n^2+n)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.3572966225745094270279188015952797... . - Vaclav Kotesovec, Sep 07 2014

A316353 Number of partitions of positive integer n such that all parts are less than the square root of n.

Original entry on oeis.org

0, 1, 1, 1, 3, 4, 4, 5, 5, 14, 16, 19, 21, 24, 27, 30, 72, 84, 94, 108, 120, 136, 150, 169, 185, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 18138, 19928, 21873, 23961, 26226, 28652

Offset: 1

Richard Locke Peterson, Jun 29 2018

nonn

This sequence itself is not a semigroup, but the set of all the partitions enumerated by this sequence does form a semigroup (actually a subsemigroup of the set of all partitions) with the following binary operation: let alpha = the partition (a,b,c,... [this is of course a finite list]) be the partition of the number N1 [that is, a + b + c + ... = N1] and let ALPHA = (A,B,C,...) be the partition of N2. Then the binary operation given by alpha*ALPHA = (a,b,c,...)*(A,B,C,...) = (aA,aB,aC,...,bA,bB,bC,...,cA,cB,cC,...) is a partition of the integer N1*N2. Furthermore, since any part x of alpha is less than the square root of N1, and likewise for any part Y of ALPHA, then the part xY is less than the square root of N1*N2, so the set is a subsemigroup of the semigroup of all partitions under the given operation. If the sole partition (1) of 1 is adjoined, the semigroup becomes a monoid.

			a(3)=1, since the partition (1,1,1) is the only partition of 3 with all parts less than the square root of 3 ~ 1.73.
a(6)=4, since there are only 4 allowable partitions: (1,1,1,1,1,1,1), (1,1,1,1,2), (1,1,2,2), and (2,2,2).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 123 terms from Robert G. Wilson v)

Cf. A000041 (the partition numbers), A097356 (with 'no greater' rather than less).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
a:= n-> b(n, (r-> `if`(r*r>=n, r-1, r))(isqrt(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 02 2018

Table[With[{s = Sqrt@ n}, Count[IntegerPartitions[n], ?(AllTrue[#, # < s &] &)]], {n, 53}] (* _Michael De Vlieger, Jul 22 2018 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt[n - 1]]; Array[f, 50] (* Robert G. Wilson v, Jul 24 2018 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := b[n, Function[r, If[r*r >= n, r - 1, r]][Floor[Sqrt[n]]]];
Array[a, 100] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

a(n) = my(nb = 0); forpart(p=n, nb++, sqrtint(n)-issquare(n)); nb; \\ Michel Marcus, Jul 15 2018

log(a(n)) ~ log(A258268) * sqrt(n) - log(n). - Vaclav Kotesovec, May 30 2021

More terms from Michel Marcus, Jul 15 2018

cp's OEIS Frontend

A097356 Number of partitions of n into parts not greater than sqrt(n).

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A206226 Number of partitions of n^2 into parts not greater than n.

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A206240 Number of partitions of n^2-n into parts not greater than n.

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A258234 Decimal expansion of a constant related to A107379.

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A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)).

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Maple

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A161407 Number of partitions of n^2 into parts smaller than n.

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A206227 Number of partitions of n^2+n into parts not greater than n.

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Maple

A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2k)(1-x^k)).