A206226 -id:A206226 - cp's OEIS frontend

A258268 Decimal expansion of a constant related to A206226.

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

Offset: 1

Vaclav Kotesovec, May 25 2015

			9.153370192454122461948530292401354540073...

Cf. A206226, A206227, A206240, A258234, A107379, A173519, A238016, A258789.

r^3/(r-1) /.FindRoot[-PolyLog[2, 1-r] == 3*Log[r]^2/2, {r, E}, WorkingPrecision->120] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A206226(n)^(1/n).
Equals limit n->infinity A206227(n)^(1/n).
Equals limit n->infinity A206240(n)^(1/n).

More digits from Vaclav Kotesovec, Jun 10 2015

A238016 Number A(n,k) of partitions of n^k into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 12, 5, 1, 1, 1, 9, 75, 64, 7, 1, 1, 1, 17, 588, 2280, 377, 11, 1, 1, 1, 33, 5043, 123464, 106852, 2432, 15, 1, 1, 1, 65, 44652, 7566280, 55567352, 6889527, 16475, 22, 1

Offset: 0

Alois P. Heinz, Feb 17 2014

In general, for k>3, is column k asymptotic to exp(2*n) * n^((k-2)*n-k) / (2*Pi). For k=1 see A000041, for k=2 see A206226 and for k=3 see A238608. - Vaclav Kotesovec, May 25 2015

Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). See also A237998, A238000, A236810 or A258668-A258672. - Vaclav Kotesovec, Jun 07 2015

			A(3,1) = 3: 3, 21, 111.
A(3,2) = 12: 333, 3222, 3321, 22221, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111.
A(2,3) = 5: 2222, 22211, 221111, 2111111, 11111111.
A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
Square array A(n,k) begins:
  0, 1,   1,      1,        1,           1, ...
  1, 1,   1,      1,        1,           1, ...
  1, 2,   3,      5,        9,          17, ...
  1, 3,  12,     75,      588,        5043, ...
  1, 5,  64,   2280,   123464,     7566280, ...
  1, 7, 377, 106852, 55567352, 33432635477, ...

Alois P. Heinz, Antidiagonals n = 0..54, flattened
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) arXiv:1108.4391 [math.CO], 2011.

Columns k=0-10 give: A057427, A000041, A206226, A238608, A238609, A238610, A238611, A238612, A238613, A238614, A238615.
Rows n=0-10 give: A057427, A000012, A094373, A238630, A238631, A238632, A238633, A238634, A238635, A238636, A238637.
Main diagonal gives A238000.
Cf. A238010.

A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, n^k}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Oct 11 2015 *)

A(n,k) = [x^(n^k)] Product_{j=1..n} 1/(1-x^j).

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

Original entry on oeis.org

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

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

A258293 Number of partitions of 3*n^2 into parts that are at most n.

Original entry on oeis.org

1, 1, 7, 75, 1033, 16019, 269005, 4767088, 87914929, 1671580383, 32560379840, 646795901962, 13058489343812, 267268692575830, 5534279506641422, 115754904055926892, 2442438538492842691, 51934447672016653655, 1111872048730513043539, 23949840661000275507964

Offset: 0

Vaclav Kotesovec, May 25 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..274

Cf. A206226, A258296, A258294, A258295.

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

(* A program to compute the constant d = 23.98280768... *) With[{j=3}, r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->100]] (* Vaclav Kotesovec, Jun 10 2015 *)

a(n) ~ c * d^n / n^2, where d = 23.98280768122086592445663786762351573848..., c = 0.0530017980244665552354063060738409813... .

A258294 Number of partitions of 4*n^2 into parts that are at most n.

Original entry on oeis.org

1, 1, 9, 127, 2280, 46262, 1015691, 23541165, 567852809, 14123231487, 359874480333, 9351900623083, 247006639629275, 6613877399621729, 179171447281396640, 4902895256737984134, 135346525073067516814, 3765244155890019687101, 105465364199865165010867

Offset: 0

Vaclav Kotesovec, May 25 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..247

Cf. A206226, A258296, A258293, A258295.

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

(* A program to compute the constant d = 31.37931997... *) With[{j=4}, r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->100]] (* Vaclav Kotesovec, Jun 10 2015 *)

a(n) ~ c * d^n / n^2, where d = 31.379319973863251370746442877119704410889..., c = 0.0397666338404544208556554596295683858... .

A258295 Number of partitions of 5*n^2 into parts that are at most n.

Original entry on oeis.org

1, 1, 11, 192, 4263, 106852, 2897747, 82966258, 2472338185, 75966810293, 2391508958235, 76782438832425, 2505642670439980, 82893573492724961, 2774547946438608789, 93807671621922558215, 3199617653993448321146, 109979504522862990517172, 3806257106793028952525938

Offset: 0

Vaclav Kotesovec, May 25 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Cf. A206226, A258296, A258293, A258294.

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

(* A program to compute the constant d = 38.7729855... *) With[{j=5}, r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->100]] (* Vaclav Kotesovec, Jun 10 2015 *)

a(n) ~ c * d^n / n^2, where d = 38.7729855097144987072847461256815071909..., c = 0.0318193213988281353709268311928... .

A258296 Number of partitions of 2*n^2 into parts that are at most n.

Original entry on oeis.org

1, 1, 5, 37, 351, 3765, 43752, 536375, 6842599, 89984614, 1212199424, 16651935901, 232477235048, 3290090540717, 47106320777132, 681247106742555, 9938641464083052, 146113228303254020, 2162784490438698636, 32209221982817148364, 482304350308369699381

Offset: 0

Vaclav Kotesovec, May 25 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..307

Cf. A206226, A258293, A258294, A258295.

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

(* A program to compute the constant d = 16.5796212... *) With[{j=2}, r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->100]] (* Vaclav Kotesovec, Jun 10 2015 *)

a(n) ~ c * d^n / n^2, where d = 16.57962120993269533568313969522872808998..., c = 0.07942450354657307077058855728600800998... .

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

A321139 a(n) = [x^(n^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

Original entry on oeis.org

1, 1, 1, 3, 7, 17, 52, 144, 480, 1732, 5902, 21078, 78434, 289107, 1079949, 4094643, 15574377, 59667023, 230318968, 892694240, 3477119540, 13606993083, 53438614380, 210622413188, 832922044686, 3303392730698, 13137474884294, 52381331536536, 209340904575968

Offset: 0

Seiichi Manyama, Oct 28 2018

nonn

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^2.

			1*0^2 + 2*1^2 + 3*1^2 + 4*0^2 + 5*2^2 = 25.
1*0^2 + 2*2^2 + 3*2^2 + 4*0^2 + 5*1^2 = 25.
1*0^2 + 2*3^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
1*1^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*2^2 = 25.
1*1^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*1^2 = 25.
1*1^2 + 2*2^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
1*1^2 + 2*2^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
1*2^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*1^2 = 25.
1*2^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*1^2 = 25.
1*2^2 + 2*1^2 + 3*1^2 + 4*2^2 + 5*0^2 = 25.
1*2^2 + 2*3^2 + 3*1^2 + 4*0^2 + 5*0^2 = 25.
1*3^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
1*3^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
1*3^2 + 2*2^2 + 3*1^2 + 4*0^2 + 5*1^2 = 25.
1*4^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*1^2 = 25.
1*4^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
1*5^2 + 2*0^2 + 3*0^2 + 4*0^2 + 5*0^2 = 25.
So a(5) = 17.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 101 terms from Seiichi Manyama)

Cf. A000122, A010052, A206226, A300446, A320932.

b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 28 2018

nmax = 25; Table[SeriesCoefficient[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

{a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}

a(n) = [x^(n^2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.

cp's OEIS Frontend

A258268 Decimal expansion of a constant related to A206226.

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A238016 Number A(n,k) of partitions of n^k into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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

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A258293 Number of partitions of 3*n^2 into parts that are at most n.

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A258294 Number of partitions of 4*n^2 into parts that are at most n.

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A258295 Number of partitions of 5*n^2 into parts that are at most n.

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A258296 Number of partitions of 2*n^2 into parts that are at most n.

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