A037444 -id:A037444 - cp's OEIS frontend

A259794 Number of partitions of n^5 into fifth powers.

1, 1, 2, 9, 88, 1104, 15772, 241582, 3869852, 63689650, 1065023018, 17948615155, 303219868652, 5116273886322, 86004191773864, 1437703756689091, 23869446608034827, 393225674878151704, 6423761195925513669, 104014146020398166139, 1668870762057827073994

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..26
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

A row of the array in A259799.

Cf. A037444, A259792, A259793, A259795.

a(n) = [x^(n^5)] Product_{k>=1} 1/(1 - x^(k^5)). - Ilya Gutkovskiy, Jan 29 2018

More terms from Alois P. Heinz, Jul 10 2015

A259795 Number of partitions of n^6 into sixth powers.

Original entry on oeis.org

1, 1, 2, 13, 218, 5082, 140549, 4318937, 142230196, 4899174096, 173759831765, 6275666535508, 229029623773326, 8400986623582879, 308552577080828413, 11315799255444002331, 413526899811283611529, 15035521464541449037361, 543292158617220114038102, 19493012206795963934830852, 694033371089826655280448205, 24509565267500567956406898725

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

A row of the array in A259799.

Cf. A037444, A259792, A259793, A259794.

a(n) = [x^(n^6)] Product_{k>=1} 1/(1 - x^(k^6)). - Ilya Gutkovskiy, Jan 29 2018

a(8)-a(14) from Alois P. Heinz, Jul 10 2015
a(15)-a(18) from Hiroaki Yamanouchi, Jul 11 2015
a(19)-a(21) from Vaclav Kotesovec, Dec 10 2016

A307608 Number of partitions of n^2 into consecutive positive squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			29^2 = 20^2 + 21^2, so a(29) = 2.

Index entries for sequences related to sums of squares

Cf. A000290, A030273, A034705, A037444, A097812, A151557, A296338.

a(n) = [x^(n^2)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^2).
a(n) = A296338(A000290(n)).
a(n) >= 2 for n in A097812.

A072925 Probably an erroneous version of A002845.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 888, 1944

Offset: 1

dead

The old entry with this sequence number was a duplicate of A037444.

J. Q. Longyear, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995; entry M1139.

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
J. Longyear, T. Trotter, N. J. A. Sloane, Correspondence

A229541 Number T(n,k) of partitions of n^2 into squares with each number of parts k; irregular triangle T(n,k), 1 <= k <= n^2.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 4, 1, 1, 4, 2, 1, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Christopher Hunt Gribble, Sep 25 2013

Row sums give A037444.

			The irregular triangle begins:
\ k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
n
1    1
2    1  0  0  1
3    1  0  1  0  0  1  0  0  1
4    1  0  0  1  1  0  1  1  0  1  0  0  1  0  0  1
5    1  0  1  1  1  2  1  1  2  1  0  1  1  0  1  1  0  1 ...
6    1  0  1  2  1  4  1  1  4  2  1  4  2  1  2  1  1  2 ...
7    1  0  1  2  2  3  4  5  3  6  6  2  5  5  2  5  4  2 ...
8    1  0  0  1  5  2  7  9  5 11  8  5 12  8  6 12  8  6 ...
9    1  0  3  2  2 10  9  9 16 16 14 17 16 14 19 18 13 20 ...
Length of row n is n^2.
For n = 3, the 4 partitions are:
Square side 1 2 3    Number of Parts
            9 0 0           9
            5 1 0           6
            1 2 0           3
            0 0 1           1
As each partition has a different number of parts,
T(3,1) = 1, T(3,3) = 1, T(3,6) = 1, T(3,9) = 1.

Christopher Hunt Gribble, Rows 1..21 flattened
Christopher Hunt Gribble, C++ program

Cf. A037444.

It appears that T(n+1,g(n+1):(n+1)^2) = T(n,f(n):n^2) where f(1) = 1, f(2) = 1, f(n) = Sum(floor(n/2)), n >= 3, g(2) = 4, g(3) = 6, g(n) = Sum(floor((n+3)/2)) + 5, n >= 4. In addition, g(n+1) - f(n) = 2n + 1 for all n.

A229961 T(n,k) is the number of partitions in each run k of strictly increasing numbers of 2 X 2 squares in the list of partitions of n^2 into squares, where partition sorting order is ascending with larger squares taking higher precedence; irregular triangle T(n,k), 1 <= n, 1 <= k <= A227940(n), read by rows.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 1, 7, 5, 2, 3, 1, 1, 10, 7, 5, 3, 1, 6, 3, 1, 2, 3, 1, 1, 13, 11, 8, 6, 4, 2, 9, 7, 4, 2, 5, 3, 1, 7, 4, 2, 3, 4, 2, 1, 17, 14, 12, 10, 8, 5, 3, 1, 13, 10, 8, 6, 4, 1, 9, 6, 4, 2, 5, 2, 1, 10, 8, 6, 4, 1, 6, 4, 2, 2, 4, 2, 8, 5, 3, 1, 4, 1, 1, 4, 2, 1

Offset: 1

Christopher Hunt Gribble, Oct 04 2013

Row lengths are given by A227940.

			For n = 4, the 8 partitions of 16 into square parts are:
Partition  Square side
.           1  2  3  4
.
.    1     16  0  0  0
.    2     12  1  0  0
.    3      8  2  0  0
.    4      4  3  0  0
.    5      0  4  0  0
.    6      7  0  1  0
.    7      3  1  1  0
.    8      0  0  0  1
So T(4,1) = 5 as the first runs of 2 X 2 squares is (0,1,2,3,4) from partitions 1 to 5;
T(4,2) = 2 as the second run is (0,1) from partitions 6 to 7;
T(4,3) = 1 as the third run is (0) from partition 8.
The irregular triangle begins:
  \  k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  n
  1     1
  2     2
  3     3  1
  4     5  2  1
  5     7  5  2  3  1  1
  6    10  7  5  3  1  6  3  1  2  3  1  1
  7    13 11  8  6  4  2  9  7  4  2  5  3  1  7  4  2  3  4 ...
  8    17 14 12 10  8  5  3  1 13 10  8  6  4  1  9  6  4  2 ...
  9    21 19 16 14 12 10  7  5  3  1 17 15 12 10  8  6  3  1 ...
  10   26 23 21 19 17 14 12 10  8  5  3  1 22 19 17 15 13 10 ...

Christopher Hunt Gribble, C++ program

Row sums = A037444.

Cf. A227940.

A321183 a(n) = [x^((n(n+1)/2)^2)] Product_{k=1..n} Sum_{m>=0} x^(k^2m).

Original entry on oeis.org

1, 1, 3, 26, 438, 11674, 434613, 21040885, 1263748763, 91057116368, 7676892453542, 742890018054927, 81267790173334794, 9926903213704358577, 1340280764681712515084, 198320073897808037293388, 31929177807445245255119558, 5558580993355817894674501169, 1040777481846356463369367882750

Offset: 0

Seiichi Manyama, Oct 29 2018

nonn

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation 1^2*a_1 + 2^2*a_2 + ... + n^2*a_n = (n*(n+1)/2)^2.

			1^2* 0 + 2^2*0 + 3^2*4 = 36.
1^2* 0 + 2^2*9 + 3^2*0 = 36.
1^2* 1 + 2^2*2 + 3^2*3 = 36.
1^2* 2 + 2^2*4 + 3^2*2 = 36.
1^2* 3 + 2^2*6 + 3^2*1 = 36.
1^2* 4 + 2^2*8 + 3^2*0 = 36.
1^2* 5 + 2^2*1 + 3^2*3 = 36.
1^2* 6 + 2^2*3 + 3^2*2 = 36.
1^2* 7 + 2^2*5 + 3^2*1 = 36.
1^2* 8 + 2^2*7 + 3^2*0 = 36.
1^2* 9 + 2^2*0 + 3^2*3 = 36.
1^2*10 + 2^2*2 + 3^2*2 = 36.
1^2*11 + 2^2*4 + 3^2*1 = 36.
1^2*12 + 2^2*6 + 3^2*0 = 36.
1^2*14 + 2^2*1 + 3^2*2 = 36.
1^2*15 + 2^2*3 + 3^2*1 = 36.
1^2*16 + 2^2*5 + 3^2*0 = 36.
1^2*18 + 2^2*0 + 3^2*2 = 36.
1^2*19 + 2^2*2 + 3^2*1 = 36.
1^2*20 + 2^2*4 + 3^2*0 = 36.
1^2*23 + 2^2*1 + 3^2*1 = 36.
1^2*24 + 2^2*3 + 3^2*0 = 36.
1^2*27 + 2^2*0 + 3^2*1 = 36.
1^2*28 + 2^2*2 + 3^2*0 = 36.
1^2*32 + 2^2*1 + 3^2*0 = 36.
1^2*36 + 2^2*0 + 3^2*0 = 36.
So a(3) = 26.

Cf. A000537, A037444, A321181, A321186.

a(16)-a(18) from Alois P. Heinz, Oct 29 2018

A092179 Number of partitions of n^2 into squares providing no dissections of the square n X n into smaller squares.

Original entry on oeis.org

0, 0, 1, 1, 8, 12, 41, 72, 192, 362, 909, 1524, 3699, 6928, 14235, 25296

Offset: 1

Reinhard Zumkeller, Apr 02 2004

a(n) = A037444(n) - A034295(n).

a(9)-a(12) from Alois P. Heinz, Apr 15 2013
a(13) from Alois P. Heinz, May 30 2013
a(14) from Christopher Hunt Gribble, Oct 26 2013
Values a(15)-a(16) computed using A034295 from Vaclav Kotesovec, Dec 07 2016

A276557 Number of partitions of prime(n)^2 into squares of primes.

Original entry on oeis.org

1, 1, 2, 4, 17, 39, 191, 410, 1771, 13805, 26459, 170897, 556698, 988053, 3019206, 15074481, 70202708, 115639004, 498047289, 1281427052, 2039282754, 7981334946, 19374343049, 71015123687, 380553620426, 862797574415, 1292837481584, 2875949125749, 4270259833946, 9334145396729

Offset: 1

Ilya Gutkovskiy, Jun 14 2017

nonn

			a(3) = 2 because third square of prime is 25 and we have [25], [9, 4, 4, 4, 4].

Cf. A001248, A037444, A056768, A078137, A090677.

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

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

a(n) = A090677(A001248(n)).

A294071 Number of ordered ways of writing n^2 as a sum of n squares > 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 6, 7, 288, 262, 13702, 69531, 610567, 5356091, 51724960, 521956086, 5467658641, 59931636545, 690518644584, 8100858045744, 99142980567486, 1246972499954475, 16142015005905558, 215722810653380845, 2955759897694815985, 41614888439136252691

Offset: 0

Ilya Gutkovskiy, Feb 07 2018

nonn

			a(5) = 5 because we have [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A037444, A066535, A078134, A092362, A232173, A280129, A280542, A281154, A281155, A298329, A298330, A298640, A298642.

Table[SeriesCoefficient[((-1 - 2 x + EllipticTheta[3, 0, x])/2)^n, {x, 0, n^2}], {n, 0, 25}]

a(n) = [x^(n^2)] (Sum_{k>=2} x^(k^2))^n.

cp's OEIS Frontend

A259794 Number of partitions of n^5 into fifth powers.

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A259795 Number of partitions of n^6 into sixth powers.

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A307608 Number of partitions of n^2 into consecutive positive squares.

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A072925 Probably an erroneous version of A002845.

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A229541 Number T(n,k) of partitions of n^2 into squares with each number of parts k; irregular triangle T(n,k), 1 <= k <= n^2.

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A321183 a(n) = [x^((n*(n+1)/2)^2)] Product_{k=1..n} Sum_{m>=0} x^(k^2*m).

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A092179 Number of partitions of n^2 into squares providing no dissections of the square n X n into smaller squares.

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A276557 Number of partitions of prime(n)^2 into squares of primes.

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A294071 Number of ordered ways of writing n^2 as a sum of n squares > 1.

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A321183 a(n) = [x^((n(n+1)/2)^2)] Product_{k=1..n} Sum_{m>=0} x^(k^2m).